Skip to main content

Full text of "A HISTORY OF MACHANICS"

See other formats


KAN'.A",  i,II  f    Mt)    IMiHUl    tWHAHY 

I 


0  ODD1  DSOSQEb 


.^*— .  * 


A  HISTORY  OF  MECHANICS 


A  HISTORY 


MECHANICS 

II  V 

HKNft  IHK;AS 

*>*;  «u*Nr£tt*;N(;»:H  <»r  tin  nun,*1,  i*t»ti 


FOKKWOttl*  in    IXM  IS  Hi-; 


J,  K.  M\UI)0\ 


w  f;mi*t4iNf  ^M^ifATKr-swtt/ 

^\,  tN<:,,  NKW  Y 


FOREWORD 


The  history  of  mechanics  is  one  of  the  most  important  branches  of 
the  history  of  science.  From  earliest  times  man  has  sought  to  develop 
tools  that  would  enable  him  to  add  to  his  power  of  action  or  to  defend 
himself  against  the  dangers  threatening  him.  Thus  he  was  uncons-* 
ciously  led  to  consider  the  problems  of  mechanics.  So  we  see  the  first 
scholars  of  ancient  times  thinking  about  these  problems  and  arriving 
more  or  less  successfully  at  a  solution.  The  motion  of  the  stars  which, 
from  the  Chaldean  shepherds  to  the  great  Greek  and  Hellenistic  astronomers, 
was  one  of  the  first  preoccupations  of  human  thought,  led  to  the  discovery 
of  the  true  laws  of  dynamics.  As  is  well  known,  although  the  principles 
of  statics  had  been  correctly  presented  by  the  old  scholars  those  of  dynamics, 
obscured  by  the  false  conceptions  of  the  aristotelian  school,  did  not  begin 
to  see  light  until  the  end  of  the  Middle  Ages  and  the  beginning  of  the  modern 
era.  Then  came  the  rapid  development  of  mechanics  due  to  the  memor* 
able  work  of  Kepler,  Galileo,  Descartes,  Huyghens  and  Newton;  the 
codification  of  its  laws  by  such  men  as  Euler,  Lagrange  and  Laplace ; 
and  the  tremendous  development  of  its  various  branches  and  the  endlessly 
increasing  number  of  applications  in  the  Nineteenth  and  Twentieth 
Centuries.  The  principles  of  mechanics  ivere  brought  to  such  a  high 
degree  of  perfection  that  fifty  years  ago  it  was  believed  that  their  develop 
ment  was  practically  complete.  It  was  then  that  there  appeared^  in  sue- 
cession^  two  very  unexpected  developments  of  classical  mechanics— on  the 
one  hand,  relativistic  mechanics  and  on  the,  other,  quantum  and  wave 
mechanics.  These  originated  in  the  necessity  of  interpreting  the  very 
delicate  phenomena  of  electromagnetism  or  of  explaining  the  observable 
processes  on  the  atomic  scale.  Whereas  reativistic  mechanics,  while  up 
setting  our  usual  notions  of  time  and  space  only*,  in  a  sense,  completed 
and  crowned  the  work  of  classical  mechanics^  the  quantum  and  wave 
mechanics  brought  us  more  radically  new  ideas  and  forced  us  to  give 
up  the  continuiuty  and  absolute  determinism  of  elementary  phenomena. 


8  HISTORY   OF   MECHANICS 

Relativistic  and  quantum  mechanics  today  form  the  two  highest  peaks  of  the 
progress  of  our  knowledge  in  the  whole  field  of  mechanical  phenomena. 
To  appraise  the  evolution  of  mechanics  from  its  origin  up  to  the  present 
time  would  be  obviously  a  difficult  task  demanding  a  considerable  amount 
of  work  and  thought.  Few  men  would  be  tempted  to  write  such  a  history 
of  mechanics ;  for  its  compilation  would  require  not  only  a  wide  and 
thorough  knowledge  of  all  the  branches  of  mechanics  ancient  and  modern, 
but  also  a  great  patience,  a  well-informed  scholarship  and  an  acute  and 
critical  mind.  These  varied  qualities  M.  Rene  Dugas — who  has  already 
become  known  for  his  fine  studies  on  certain  particular  themes  in  the 
history  of  dynamics  and  for  his  critical  essays  on  different  matters  in  class 
ical,  relativistic  and  quantum  mechanics — unites  these  to  a  high  degree. 
More  than  this,  he  has  tackled  this  overwhelming  task  and,  after  several 
years,  has  brought  it  to  a  successful  conclusion.  The  important  work 
that  he  now  publishes  on  the  history  of  mechanics  constitutes  a  compre 
hensive  view  of  the  greatest  interest  which  will  be  highly  appreciated  by 
all  those  who  study  the  history  of  scientific  thought. 

Mr.  Dugas'  book  is  in  certain  ways  comparable  with  Ernest  Mack's 
famous  book  "  Mechanics,  A  historical  and  critical  presentation  of  its 
development.  "  Certainly  the  reading  of  Mach's  book,  so  full  of  original 
ideas  and  profound  comments,  is  still  extremely  instructive  and  absorbing. 
But  Mr.  Dugas''  history  of  mechanics  has  the  advantage  of  being  less 
systematic  and  more  complete.  Mactfs  thought  was  in  fact  dominated 
by  the  general  ideas  which  secured  his  adherence  in  Physics  to  the  energetic 
school  and  in  Philosophy  to  the  positivistic  thesis.  He  frequently  sought 
to  find  an  illustration,  in  the  history  of  mechanics,  of  his  own  ideas.  Often 
this  gives  his  book  a  character  which  is  a  little  too  systematic,  that  of  a 
thesis  in  which  the  arguments  in  favour  of  preconceived  ideas  are 
rehearsed.  Mr.  Dugas'  attitude  is  quite  different.  A  scrupulous  historian, 
he  has  patiently  followed  all  the  vagaries  of  thought  of  the  great  students 
of  the  subject,  collating  their  texts  carefully  and  always  preserving  the 
strictest  objectivity. 

More  impartial  than  Mach,  Mr.  Dugas  has  been  helped  by  the,  deve 
lopment  of  historical  criticism  on  the  one  hand,  by  the  progress  of  science 
on  the  other,  and  has  been  able  to  be  more  complete.  He  has  given  us 
a  much  more  detailed  picture  of  the  efforts  that  were  made  and  the  remits 
obtained  in  Antiquity  and,  especially,  in  the  Middle  Ages.  It  is  parti 
cularly  to  the  authoritative  ivork  of  Pierre  Duhem  that  Mr.  Dugas  oiws 


FOREWORD  9 

his  ability  to  show  us  the  important  contributions  made  to  the  development 
of  the  principles  of  mechanics  by  masters  like  Jordanus  of  IVemore,  Jean 
Buridan,  Albert  of  Saxony ,  Nicole  Oresme  and  a  great  artist  of  universal 
interest  like  Leonardo  da  Vinci.  Of  DuhemSs  magnificent  researches 
— which  are  often  a  little  difficult  to  study  in  that  eminent  and  erudite  phy- 
sicisfs  original  text,  usually  lengthy  and  somewhat  vague — Mr.  Dugas 
has  been  able  to  make,  in  a  few  pages,  a  short  presentation  that  the  reader 
will  read  easily  and  tvith  the  greatest  profit. 

Well  informed  of  the  most  recent  progress  of  the  science,  the  author, 
accustomed  to  reflect  of  the  new  contemporary  forms  of  mechanics,  has 
devoted  the  last  part  of  his  book  to  relativistic  mechanics  and  wave  and 
quantum  mechanics.  This  very  accurate  presentation  made  by  following 
closely,  as  is  the  authors  practice,  the  ideas  of  the  innovators  and  the 
text  of  their  writings,  naturally  makes  Mr.  Dugas'  history  of  mechanics 
much  more  complete  that  those  of  his  predecessors. 

The  central  part  of  the  book,  devoted  to  the  developments  of  mechanics 
in  the  Seventeenth,  Eighteenth  and  Nineteenth  Centuries,  has  demanded 
a  great  amount  of  work^  for  the  material  is  immense.  Being  unable  to 
follow  all  the  details  of  the  development  of  mechanics  in  the  Eighteenth 
Century,  and  especially  in  the  Nineteenth  Century,  Mr.  Dugas  has  selected 
for  a  thorough  study  certain  questions  of  special  importance,  either  in 
themselves  or  because  of  the  extensions  which  they  hav&  had  into  the  con* 
temporary  period.  It  seems  to  me  that  this  selection  has  been  made  very 
skillfully  and  has  enabled  the  author,  without  losing  himself  in  details, 
to  outline  the  principal  paths  followed  by  scientific  thought  in  this  domain. 

Perhaps,  in,  reading  Mr.  Dugas''  so  clear  text,  the  reader  will  not 
appreciate  the  work  that  the  writing  of  such  a  book  represents.  Not 
only  has  Mr.  Dugas  had  to  sift  various  questions  to  select  those  which 
would  most  clearly  illustrate  the  decisive  turning-points  in  the  progress 
of  mechanics,  but  he  has  always  referred  to  the  original  texts  themselves, 
never  wanting  to  accomplish  the  task  at  second  hand.  When^  for  ex« 
ample  ^  he  summarises  for  us  the  work  of  Kepler  in  a  few  p&g®s>>  it  is 
after  having  re-examined  and*  in  some  way,  rethought  these  arguments 
—often  complicated  and  a  tittle  quaint  and,  moreover,  written  in  a  bad 
lAitin  whose  meaning  is  often  difficult  to  appreciate—which  enabled 
the  great  astronomer  to  discover  the  correct  laws  of  the  motion  of  the  planets. 
It  is  this  necessary  conjunction  of  the  procedures  of  a  patient  erudition  and  a 
wide  knowledge  of  the  past  and  present  results  of  the  science  which  mak$s 


10  HISTORY   OF   MECHANICS 

the  history  of  science  particularly  difficult  and  restricts  the  number  of 
those  who  can,  with  profit,  devote  themselves  to  it.  Therefore  Mr.  Rene 
Dugas  should  be  warmly  thanked  for  having  placed  at  the  service  of  the 
history  of  science,  qualities  of  mind  and  methods  of  work  rarely  united 
in  one  man,  and  for  having  given  us  a  remarkable  work  which  will  remain 
a  document  of  the  first  rank  for  the  historian  of  mechanics. 


of  the  AcadAmie  franyaite 
permanent  secretary  of  the  Academic  des  sciences , 


PREFACE 


Mechanics  is  one  of  the  branches  of  physics  in  "which  the  number 
of  principles  is  at  once  very  few  and  very  rich  in  useful  consequences. 
On  the  other  hand,  there  are  few  sciences  which  have  required  so 
much  thought —  the  conquest  of  a  few  axioms  has  taken  more  than 
2000  years. 

As  Mr.  Joseph  Per^s  has  remarked,  to  speak  of  the  miracle  of  Greece 
or  of  the  night  of  the  Middle  Ages  in  the  evolution  of  mechanics  is  not 
possible.  Correctly  speaking,  Archimedes  was  able  to  conquer  statics 
and  knew  how  to  construct  a  rational  science  in  which  the  precise 
deductions  of  mathematical  analysis  played  a  part.  But  hellenic 
dynamics  is  now  seen  to  be  quite  erroneous.  It  was  however,  in  touch 
with  every-day  observation.  But,  being  unable  to  recognise  the  func 
tion  of  passive  resistances  and  lacking  a  precise  kinematics  of  accelerated 
motion,  it  could  not  serve  as  a  foundation  for  classical  mechanics. 

The  prejudices  of  the  Schoolmen,  whose  authority  in  other  fields 
was  undisputed,  restricted  the  progress  of  mechanics  for  a  long  period. 
Annotating  Aristotle  wan  the  essential  purpose  of  teaching  throughout 
the  Middle  Ages.  Not  that  the  mediaeval  scholars  lacked  originality. 
Indeed,  they  displayed  an  acute  subtlety  which  haa  never  been  surpassed, 
but  most  often  they  neglected  to  take  account  of  observation,  preferring 
to  exercise  their  minds  in  a  pure  field.  Only  the  astronomers  were  an 
exception  and  accumulated  the  facts  on  which,  much  later,  mechanics 
was  to  be  based* 

The  Thirteenth  Century  had,  however,  an  original  school  of  ntatics 
which  advocated,  in  the  treatment  of  heavy  bodies,  a  new  principle 
— under  the  name  of  gmvitax  secundum  situm— that  was  to  develop 
into  the  principle  of  virtual  work;  moreover,  thitf  principle  solved,  long 
before  Stevin  and  Galileo,  the  problem  of  tin*  equilibrium,  of  a  heavy 
body  on  an  inclined  plane,  which  Pappus  had  not  sueeeecled  in  doing 
correctly.  In  the  fourteenth  century,  Buridan  formulated  the  first 
theory  of  energy  under  the  name  of  impetus.  Thin  theory  explicitly 
departs  from  the  Peripatetic  ideas,  which  demanded  the  constant  inter 
vention  of  a  mover  to  maintain  violent  motion  in  the  Aristotelian  sense. 
Incorporated  into  a  continued  tradition  in  which  it  was  deformed  in 


12  HISTORY   OF   MECHANICS 

order  to  conform  to  an  animist  doctrine,  which  in  the  hands  of  the 
German  metaphysicians  of  the  fifteenth  century  was  to  subsist  with 
Kepler,  the  theory  of  impetus  became,  in  the  hands  of  Benedetti,  an 
early  form  of  the  principle  of  inertia,  while  one  of  its  other  aspects  was 
to  become,  after  a  long  polemic,  the  doctrine  of  vis  viva.  And  in  the 
Fourteenth  Century,  the  Oxford  School,  which  in  other  respects  indulged 
in  such  artificial  quibbling,  was  to  clarify  the  laws  of  the  kinematics  of 
uniformly  accelerated  motion. 

The  mechanics  of  the  Middle  Ages  received  something  of  a  check 
during  the  Renaissance,  which  caused  a  return  to  classical  traditions. 
The  Schools  were  attacked  by  the  humanists.  Yet,  before  Galileo, 
Dominico  Soto  successfully  formulated  the  exact  laws  of  heavy  bodies 
even  if  he  did  not  verify  them  experimentally. 

Under  what  may  seem  an  ambitious  title,  A  History  of  Mechanics, 
we  shall  deal  with  the  evolution  of  the  principles  of  general  mechanics, 
while  we  shall  omit  the  practical  applications  and,  a  fortiori,  tech 
nology. 

As  far  as  possible  we  shall  follow  a  chronological  order,  in  the  manner 
of  elementary  text  books  which  begin  with  early  history  and  end  with 
the  latest  war.  After  considerable  reflexion,  this  order  has  seemed  to 
us  preferable  to  the  one  adopted  by  important  critics,  which  consists 
in  choosing  a  given  principle  or  problem  and  analysing  its  evolution 
throughout  the  centuries.  This  latter  method  offers  the  advantage  of 
isolating  a  theory  and  treating  it  closely  ;  it  lends  itself  to  short  cuts 
which  are  striking  but  which  sometimes  seem  a  little  artificial  :  the 
different  key  problems  of  mechanical  science  evolved  in  fact  along 
parallel  lines,  profiting  by  the  progress  made  in  mathematical  language  ; 
what  is  more,  these  problems  were  interconnected.  We  have  preferred 
here  to  follow  the  elementary  order  in  time.  Each  century  will  thus 
appear  in  full  light,  with  its  own  mentality  and  atmosphere.  So  we 
jump  necessarily  from  one  theme  to  another,  but  the  works  find  once 
more  their  unity  and  their  natural  background,  without  the  distortion 
caused  by  juxtaposing  them  in  time. 

The  present  book  is  divided  into  five  parts.  The  first  treats  of  the 
originators,  the  precursors,  from  the  beginning  up  to  and  including 
the  Sixteenth  Century;  the  second,  of  the  formation  of  classical 
mechanics.  In  this  domain  the  Seventeenth  Century  deserves  to  be 
considered  as  the  great  creative  century,  with  three  great  peaks  formed 
by  Galileo,  Huyghens  and  Newton.  The  third  part  is  devoted  to  the 
Eighteenth  Century,  which  emerges  as  the  century  of  the  organisation 
of  mechanics  and  finds  its  climax  in  the  work  of  Lagrange,  immediately 


PREFACE  13 

preceded  by  Euler  and  d'Alembert.  The  development  of  mathematical 
analysis  enabled  mechanics  to  take  a  form  which,  for  a  considerable 
time,  appeared  to  be  finally  established,  and  which  is  still  a  part  of 
the  classical  teaching. 

We  have  found  ourselves  somewhat  embarrassed  in  the  writing 
of  the  fourth  part,  which  is  concerned  with  classical  mechanics  after 
Lagrange.  Indeed,  nothing  would  be  gained  by  duplicating  the 
textbooks.  Therefore  we  have  confined  ourselves  to  a  selection  from 
the  work  of  the  Nineteenth  Century  and  the  beginning  of  the  Twentieth 
Century.  This  selection  may  appear  somewhat  arbitrary  to  the  well- 
informed  reader.  Moreover,  it  is  rather  interesting  to  observe  that 
the  uneasiness  about  the  structure  of  classical  mechanics  which  is 
evident  in  the  writings  of  the  critics  did  not  prepare  the  way  for  the 
relativistic  and  quantum  revolutions,  dated  1905  and  1923  respectively. 
These  came  from  outside,  imposed  by  the  necessities  of  optics,  electro- 
magnetism  and  the  theory  of  radiation.  However,  these  reflections 
of  the  critics  were  not  valueless,  for  they  showed  that  the  lagrangian 
science  was  not  intangible  and  that  the  axioms  of  mechanics,  to  use 
Mach's  words,  "  not  only  assumed  but  also  demanded  the  continual 
control  of  experiment." 

Thus  the  determinism — or,  if  it  is  preferred,  legality — of  which  the 
students  of  mechanics  were  so  proud  and  which  made  their  science 
the  model  of  all  physical  theories,  now  appears,  after  the  development 
of  wave  mechanics,  as  justified  in  the  macroscopic  domain  because  of 
statistical  compensation  between  the  individuals  of  a  large  assembly, 
without  there  being  an  underlying  determinism. 

The  fact  of  collecting  in  one  book  the  stammerings  of  the  early 
students,  the  creation  and  organisation  of  the  classical  science  and  the 
rudiments  of  the  new  mechanics— the  object  of  the  fifth  part  of  this 
book — may  appear  a  wager.  It  is  only  so  iu  appearance.  Indeed, 
the  original  works  never  had  that  codified  aspect  which  is,  of  necessity, 
lent  them  by  the  textbooks.  Just  as  the  French  currency  remained 
stable  for  more  than  a  century,  Lagrange'n  mechanics  was  able  to 
appear  as  complete  for  a  period  of  roughly  the  same  length*  Despite 
its  mathematical  perfection,  it  had  no  other  foundation  that  that  of 
common  experiment.  The  double  revolution  of  1905  and  1923— the 
second  much  more  radical  than  the  first—profoundly  disturbed  the 
classical  structure.  For  these  new  doctrines  intended  to  rule  over 
the  whole  of  physics,  only  admitting  the  validity  of  classical  mechanics 
in  the  limits  at  which  the  velocity  oflight  can  be  considered  infinite  and 
Planck's  universal  constant  negligible.  As  regards  the  principles,  cer 
tain  thinkers  have  made  the  mistake  of  incorporating  into  a  system 


14  HISTORY    OF   MECHANICS 

what  was  only  a  stability  of  fact,  a  stage  in  evolution,  as  long  as  this 
stability  appeared  to  be  verified  and  however  important  were,  and  still 
are,  its  conquests. 

For  the  historian,  who  is  only  a  spectator  and  who,  by  his  profession, 
is  aware  of  the  fragility  of  our  anthropomorphic  science,  these  revolu 
tions  are  not  an  object  of  scandal  or  even  a  surprise.  This  does  not 
mean  that  innovations  must  be  accepted  without  analysis,  or  new 
dogmas  professed  without  reserve. 

It  is  difficult  for  us  to  indicate  in  detail  what  this  work  owes  to  the 
great  historians  and  critics  of  mechanics.  From  Duhem  we  have  taken 
what  material  he  could  extract  from  the  manuscripts  of  the  Middle 
Ages,  at  the  same  time  bringing  to  this  semi-darkness  the  light  of  his 
particularly  profound  and  alert  mind.  We  must  confess  however  that 
we  have  had  to  disagree  with  several  of  his  opinions,  which  appeared 
too  categorical  to  us.  Duhem  had  undergone  the  polarisation  of  the 
investigator  which  leads  him  to  attach  too  great  an  importance  some 
times  to  the  original  he  has  just  discovered.  Besides,  Duhem's  attach 
ment  to  energetics  made  him  somewhat  biassed.  For  example,  the 
principle  of  virtual  work  triumphed  over  that  of  virtual  velocity  which, 
up  to  the  the  early  writings  of  Galileo,  remained  inspired  by  peripatetic 
doctrines,  but  it  already  explained  correctly  the  laws  of  the  equilibrium 
of  simple  machines,  and  long  preceded  the  former.  Curiously  enough, 
Duhem  makes  Leonardo  da  Vinci  the  centre  of  his  studies  of  the  Pre 
cursors,  to  the  extent  of  considering  the  Unknown  Man  of  the  Thirteenth 
Century,  that  same  one  who  discovered  the  law  of  the  equilibrium  of  a 
heavy  body  on  an  inclined  plane,  as  a  precursor  of  Leonardo,  whereas 
this  latter  was  always  in  error  on  this  problem,  and  of  qualifying  as 
plagiaries  or  successors  of  Leonardo  a  large  number  of  authors  of  the 
Sixteenth  Century  who  may  very  well  not  have  come  under  the  influence 
of  the  great  painter  who  in  mechanics  was  scarcely  more  than  an 
amateur  of  genius. 

These  few  reserves,  for  which  we  beg  to  be  excused,  do  not  prevent 
Duhem's  historical  work  from  being  of  the  greatest  importance.  An 
indefatigable  reader,  he  succeeded  not  only  in  bringing  to  life  workfi  of 
the  Scholastics  of  the  Middle  Ages,  that  were  until  then  little  known, 
but  in  establishing  between  them  and  the  classical  period  filiations  of 
indisputable  interest.  It  is  certain  that,  on  more  than  one  point,  this 
Scholastic  sheds  light  on  and  prefigures  Descartes. 

From  Emile  Jouguet,  who  honoured  us  with  his  teaching,  we  have 
borrowed  several  of  his  Lectures — given  conscientiously  and  with  great 
regard  for  the  original  authors— and  a  number  of  opinions  that  were, 


PREFACE  15 

out  of  modesty,  consigned  to  notes,  lest  they  should  hide  his  perfect 
knowledge  of  the  Ancients. 

In  many  places  we  have  cited  the  very  personal,  and  sometimes 
very  judicious,  observations  of  Mach.  His  Mechanics  was  one  of  the 
first  systematic  works  of  its  kind  and  represented  both  a  very  wide 
reading  and  a  critical  mind  of  remarkable  independence.1 

Strictly  speaking,  Painleve  did  not  treat  the  history  of  mechanics. 
On  his  own  account,  with  the  analytical  mind  that  he  applied  to  every 
thing,  he  rethought  the  evolution  of  mechanics.  The  lectures  which 
he  gave  us  at  the  £cole  pofytechnique  were  revised  and  developed  in 
his  Axiomes  de  la  Mecanique.  This  contains  not  only  an  original 
discussion  of  the  classical  principles,  but  an  often  constructive  and 
always  valuable  criticism  of  relativistic  doctrines. 

In  spite  of  the  contributions  of  the  great  critics  we  have  just  mention 
ed,  in  spite  of  the  several  researches  of  the  original  authors  themselves 
which  this  book  contains,  we  do  not  conceal  its  imperfections  and  its 
omissions.  Certain  of  these  omissions,  especially  from  the  classical 
field  after  Lagrange,  have  been  accepted  deliberately  ;  others  may  be 
unknown  to  us  and,  for  this  reason,  more  serious.  We  have  not  sought 
to  restrict  ourselves  too  narrowly  to  our  nubject,  and  have  made  some 
incursions  into  the  domain  of  astronomy  and  that  of  hydrodynamics 
when  it  seemed  that  these  served  our  purpose.  But  a  presentation 
of  a  system  of  the  world,  or  a  complete  history  of  the  mechanics  of 
fluids,  should  not  be  looked  for  here  ;  these  subjects  themselves  would 
require  whole  volumes. 

This  book  will  only  be  read  with  profit  by  those  who  already  have 
some  knowledge  of  the  didactic  aspect  of  things.  It  also  presupposes, 
as  does  all  mechanics,  a  somewhat  extensive  mathematical  background. 
Our  purpose  will  have  been  achieved  if  the  reader  finds  in  it,  with  less 
effort  than  it  has  cost  us  to  unite  and  explain  the  original  texts,  a  reflec 
tion  of  the  joy  of  knowledge  that  we  have  found. 

I  must  thank  the  "  Editions  du  Griffon  "  for  having  applied  all  their 
recourses  to  the  production  of  this  book, 

1  Throughout  this  book,  Du  HEM'S  Origines  de  ia  Statique  are  indicated  by  the 
initials  0.  6'.,  MACK'S  Mechanics  by  the  imtial  M.,  and  JOUGUKT'B  Lectures  de  Mtca- 

nique  by  the  initials  L.  M. 


PART  ONE 


THE  ORIGINS 


CHAPTER   ONE 


HELLENIC  SCIENCE 


1.  ARISTOTELIAN  MECHANICS. 

For  lack  of  more  ancient  records,  history  of  mechanics  starts  with 
Aristotle  (384-322  B.  C.)  or,  more  accurately,  with  the  author  of  the 
probably  apocryphal  treatise  called  Problems  of  Mechanics  (Mriywvmv. 
nQoftkrifjiVLTy.) .  This  is,  in  fact,  a  text-book  of  practical  mechanics 
devoted  to  the  study  of  simple  machines. 

In  this  treatise  the  power  of  the  agency  that  sets  a  body  in  motion  is 
defined  as  the  product  of  the  weight  or  the  mass  of  the  body — the 
Ancients  always  confused  these  concepts — and  the  velocity  of  the  motion 
which  the  body  acquires.  This  law  makes  it  possible  to  formulate  the 
condition  of  equilibrium  of  a  straight  lever  with  two  unequal  arms 
which  carry  unequal  weights  at  their  ends.  Indeed,  when  the  lever 
rotates  the  velocities  of  the  weights  will  be  proportional  to  the  lengths 
of  their  supporting  arms,  for  in  these  circumstances  the  powers  of  the 
two  opposing  powers  cancel  each  other  out. 

The  author  regards  the  efficacy  of  the  lever  as  a  consequence  of  a 
magical  property  of  the  circle.  "  Someone  who  would  not  be  able  to 
move  a  load  without  a  lever  can  displace  it  easily  when  he  applies  a 
lever  to  the  weight.  Now  the  root  cause  of  all  such  phenomena  is  the 
circle.  And  this  is  natural,  for  it  is  in  no  way  strange  that  something 
remarkable  should  result  from  something  which  is  more  remarkable, 
and  the  most  remarkable  fact  i«  the  combination  of  oppoeites  with  each 
other.  A  circle  is  made  up  of  such  opposites,  for  to  begin  with  it  is 
made  up  of  something  which  moves  and  something  which  remains 
stationary.  ..."  * 

In  this  way  Problems  of  Mechanics  reduces  the  study  of  all  simple 
machines  to  one  and  the  same  principle.  **  The  properties  of  the  balance 
are  related  to  those  of  the  circle  and  the  properties  of  the  lever  to  those 
of  the  balance.  Ultimately  most  of  the  motions  in  mechanics  are 
related  to  the  properties  of  a  lever.  '* 


20  THE    ORIGINS 

To  Aristotle  himself,  just  as  much  in  his  Treatise  on  the  Heavens, 
(IIsQi  OVQWOV)  as  in  his  Physics,  concepts  belonging  to  mechanics  were 
not  differentiated  from  concepts  having  a  more  general  significance. 
Thus  the  notion  of  movement  included  both  changes  of  position  and 
changes  of  kind,  of  physical  or  chemical  state.  Aristotle's  law  of 
powers,  which  he  caUed  Mvayug  or  fcyj^g,  is  formulated  in  Chapter  V 
of  Book  VII  of  his  Physics  in  the  following  way. 

"  Let  the  motive  agency  be  a,  the  moving  body  /?,  the  distance  tra 
velled  y  and  the  time  taken  by  the  displacement  be  <5.  Then  an  equal 
power,  namely  the  power  a,  will  move  half  of  /?  along  a  path  twice  y  in 
the  same  time,  or  it  will  move  it  through  the  distance  y  in  half  the  time  d. 
For  in  this  way  the  proportions  will  be  maintained. " 

Aristotle  imposed  a  simple  restriction   on  the  application  of  this 

rule— a  small  power  should  not  be  able  to  move  too  heavy  a  body, 

"  for  then  one  man  alone  would  be  sufficient  to  set  a  ship  in  motion.  " 

This  same  law  of  powers  reappears  in  Book  III  of  the  Treatise  on 

the  Heavens.     Its  application  to  statics  may  be  regarded  as  the  origin 

of  the  principle  of  virtual  velocities  which  will  be  encountered  much  later. 

In   another   place   Aristotle   made    a   distinction   between    natural 

motions  and  violent  motions. 

The  fall  of  heavy  bodies,  for  example,  is  a  natural  motion,  while  the 
motion  of  a  projectile  is  a  violent  one. 

To  each  thing  corresponds  a  natural  place.  In  this  place  its 
substantial  form  achieves  perfection — it  is  disposed  in  such  a  way  that 
it  is  subject  as  completely  as  possible  to  influences  which  are  favourable, 
and  so  that  it  avoids  those  which  are  inimical.  If  something  is  moved 
from  its  natural  place  it  tends  to  return  there,  for  everything  tends  to 
perfection.  If  it  already  occupies  its  natural  place  it  remains  there  at 
rest  and  can  only  be  torn  away  by  violence. 

In  a  precise  way,  for  Aristotle,  the  position  of  a  body  is  the  internal 
surface  of  the  bodies  which  surround  it.  To  his  most  faithful  commen 
tators,  the  natural  place  of  the  earth  is  the  concave  surface  which  deiines 
the  bottom  of  the  sea,  joined  in  part  to  the  lower  surface  of  the  atmo 
sphere,  the  natural  place  of  the  air.1 

Concerning  the  natural  motion  of  falling  bodies,  Aristotle  maintained 
in  Book  I  of  the  Treatise  on  the  Heavens  that  the  "  relation  which  weights 
have  to  each  other  is  reproduced  inversely  in  their  durations  of  fall. 
If  a  weight  falls  from  a  certain  height  in  so  much  time,  a  weight  which 
is  twice  as  great  will  fall  from  the  same  height  in  half  the  time* 

In  his  Physics  (Part  V),  Aristotle  remarked  on  the  acceleration  of 

1  Cf.  DXIHEM,  Origines  de  la  Statique,  Vol.  II,  p.  21.  Throughout  the  prenent  hook 
this  work  of  Duhem  will  be  indicated  by  the  letters  0.  S. 


HELLENIC   SCIENCE  21 

falling  heavy  bodies.  A  body  is  attracted  towards  its  natural  place  by 
means  of  its  heaviness.  The  closer  the  body  comes  to  the  ground,  the 
more  that  property  increases. 

If  the  natural  place  of  heavy  bodies  is  the  centre  of  the  World,  the  na 
tural  place  of  light  bodies  is  the  region  contiguous  with  the  Sphere  of  the 
Moon.  Heavenly  bodies  are  not  subject  to  the  laws  applicable  to  terres 
trial  ones — every  star  is  a  body  as  it  were  divine,  moved  by  its  own  divinity. 

We  return  to  terrestrial  mechanics.  All  violent  motion  is  essentially 
impermanent.  This  is  one  of  the  axioms  which  the  Schoolmen  were  to 
repeat — Nullum  violentum  potest  esse  perpetuum.  Once  a  projectile  is 
thrown,  the  motive  agency  which  assures  the  continuity  of  the  motion 
resides  in  the  air  which  has  been  set  in  motion.  Aristotle  then  assumes 
that,  in  contrast  to  solid  bodies,  air  spontaneously  preserves  the  impul 
sion  which  it  receives  when  the  projectile  is  thrown,  and  that  it  can  in 
consequence  act  as  the  motive  agency  during  the  projectile's  flight. 

This  opinion  may  seem  all  the  more  paradoxical  in  view  of  the  fact 
that  Aristotle  remarked,  elsewhere,  on  the  resistance  of  the  medium. 
This  resistance  increases  in  direct  proportion  to  the  density  of  the  me 
dium.  "  If  air  is  twice  as  tenuous  as  water,  the  same  moving  body 
will  spend  twice  as  much  time  in  travelling  a  certain  path  in  water  as 
in  travelling  the  same  path  in  air.  " 

Aristotle  also  concerned  himself  with  the  composition  of  motions. 
"  Let  a  moving  body  be  simultaneously  actuated  by  two  motions  that 
are  such  that  the  distances  travelled  in  the  same  time  are  in  a  constant 
proportion.  Then  it  will  move  along  the  diagonal  of  a  parallelogram 
which  has  as  sides  two  lines  whose  lengths  are  in  this  constant  relation 
to  each  other.  "  On  the  other  hand,  if  the  ratio  between  the  two  com 
ponent  distances  travelled  by  the  moving  body  in  the  name  time  varies 
from  one  instant  to  another,  the  body  cannot  have  a  rectilinear  motion. 
"  In  such  a  way  a  curved  path  is  generated  when  the  moving  body  is 
animated  by  two  motions  whose  proportion  does  not  remain  constant 
from  one  instant  to  another. " 

These  propositions  relate  to  what  we  now  call  kinematics.  But 
Aristotle  immediately  inferred  from  them  dynamical  results  concerning 
the  composition  of  forces.  The  connection  between  the  two  disciplines 
is  not  given,  but  as  Dutxem  has  indicated,  it  is  easily  supplied  by  making 
use  of  the  law  of  powers— a  fundamental  principle  of  aristotelian  dyna 
mics.  In  particular,  let  us  consider  a  heavy  moving  body  describing 
some  curve  in  a  vertical  plane.  It  is  clear  that  the  body  is  actuated 
by  two  motions  simultaneously.  Of  these,  one  produces  a  vertical 
descent  while  the  other,  according  to  the  position  of  the  body  on  its 
trajectory,  results  in  an  increase  or  a  decrease  of  the  distance  from  the 


22  THE    ORIGINS 

centre.  In  Aristotle's  sense,  the  body  will  have  a  natural  falling  motion 
due  to  gravity,  and  will  be  carried  horizontally  in  a  violent  motion. 
Consider  different  moving  bodies  unequally  distant  from  the  centre  of 
a  circle  and  on  the  same  radius.  Let  this  radius,  in  falling,  rotate  about 
the  centre.  Then  it  may  be  inferred  that  for  each  body  the  relation  of 
the  natural  to  the  violent  motion  remains  the  same.  "  The  contem 
plation  of  this  ecpiality  held  Aristotle's  attention  for  a  long  time.  He 
appears  to  have  seen  in  it  a  somewhat  mysterious  correlation  with  the 
law  of  the  equilibrium  of  levers.  "  x 

Aristotle  believed  in  the  impossibility  of  a  vacuum  (Physics,  Book  IV, 
Chapter  XI)  on  the  grounds  that,  in  a  vacuum,  no  natural  motion,  that 
is  to  say  no  tendency  towards  a  natural  place,  would  be  possible. 
Incidentally  this  idea  led  him  to  formulate  a  principle  analogous  to 
that  of  inertia,  and  to  justify  this  in  the  same  way  as  that  used  by  the 
great  physicists  of  the  XVIIIth  Century. 

"  It  is  impossible  to  say  why  a  body  that  has  been  set  in  motion  in 
a  vacuum  should  ever  come  to  rest ;  why,  indeed,  it  should  come  to 
rest  at  one  place  rather  than  at  another.  As  a  consequence,  it  will 
either  necessarily  stay  at  rest  or,  if  in  motion,  will  move  indefinitely 
unless  some  obstacle  comes  into  collision  with  it.  " 

Aristotle's  ideas  on  gravitation  and  the  figure  of  the  Earth  merit  our 
attention,  if  only  because  of  the  influence  which  they  have  had  on  the 
development  of  the  principles  of  mechanics.  First  we  shall  quote  from 
the  Treatise  on  the  Heavens  (Book  II,  Chapter  XIV).  "  Since  the 
centres  of  the  Universe  and  of  the  Earth  coincide,  one  should  ask  one 
self  towards  which  of  these  heavy  bodies  and  even  the  parts  of  the 
Earth  are  attracted.  Are  they  attracted  towards  this  point  because  it 
is  the  centre  of  the  Universe  or  because  it  is  the  centre  of  the  Earth  ? 
It  is  the  centre  of  the  Universe  towards  which  they  must  be  attracted. . . . 
Consequently  heavy  bodies  are  attracted  towards  the  centre  of  the 
Earth,  but  only  fortuitously,  because  this  centre  is  at  the  centre  of  the 
Universe. " 

If  the  Earth  is  spherical  and  at  the  centre  of  the  World,  what 
happens  if  a  large  weight  is  added  to  one  of  the  hemispheres  ?  The 
answer  to  this  question  is  the  following.  "  The  Earth  will  necessarily 
move  until  it  surrounds  the  centre  of  the  World  in  a  uniform  way,  tlie 
tendencies  to  motion  of  the  different  parts  counterbalancing  one  an 
other."  Duhem  points  out  that  the  centre,  TO  ueaov,  that  in  every  body 
is  attracted  to  the  centre  of  the  Universe,  was  not  defined  in  a  precise 

1  DUHEM,  0.  S.,  Vol.  I,  p.  110.  Note  here  that,/or  the  same  fall,  the  longer  the'lever 
the  less  the  natural  motion  will  be  disturbed.  It  is  therefore  natural  to  assume  that 
a  weight  has  more  power  at  the  end  of  a  long  lever  than  at  the  end  of  a  short  one. 


HELLENIC   SCIENCE  23 

way  by  Aristotle.     In  particular,  Aristotle  did  not  identify  it  with  the 
centre  of  gravity,  of  which  he  was  ignorant.1 

In  this  same  treatise  Aristotle  repeatedly  enumerates  the  arguments 
for  the  spherical  figure  of  the  Earth.  He  distinguishes  a  posteriori 
arguments,  such  as  the  shape  of  the  Earth's  shadow  in  eclipses  of  the 
Moon,  the  appearance  and  disappearance  of  constellations  to  a  traveller 
going  from  north  to  south,  from  a  priori  arguments,  of  which  he  says — 

"  Suppose  that  the  Earth  is  no  longer  a  single  mass,  but  that,  poten 
tially,  its  different  parts  are  separated  from  each  other  and  are  placed 
in  all  directions  and  attracted  similarity  towards  the  centre.  Then  let 
the  parts  of  the  Earth  which  have  been  separated  from  each  other  and 
taken  to  the  ends  of  the  World  be  allowed  to  reunite  at  the  centre  ; 
let  the  Earth  be  formed  by  a  different  procedure — the  result  will  be 
exactly  the  same.  If  the  parts  are  taken  to  the  ends  of  the  World  and 
are  taken  there  similarily  in  all  directions,  they  will  necessarily  form  a 
mass  which  is  symmetrical.  Because  there  will  result  an  addition  of 
parts  which  are  equal  in  all  directions,  and  the  surface  which  defines  the 
mass  produced  will  be  everywhere  equidistant  from  the  centre.  Such 
a  surface  will  therefore  be  a  sphere.  But  the  explanation  of  the  shape 
of  the  Earth  would  not  be  changed  in  any  way  if  the  parts  which  form 
it  were  not  taken  in  equal  quantities  in  all  directions.  In  fact,  a  larger 
part  will  necessarily  push  away  a  smaller  one  which  it  finds  in  front  of 
it,  for  both  have  a  tendency  towards  the  centre  and  more  powerful 
weights  are  able  to  displace  lesser  ones.  " 

To  Aristotle,  heaviness  does  not  prove  rigorously  that  the  Earth 
will  be  spherical,  but  only  that  it  will  tend  to  be  so.  On  the  other 
hand,  for  the  surface  of  water,  this  is  obvious  if  it  is  admitted  that  "  it 
is  a  property  of  water  to  run  towards  the  lowest 
places, "  that  is,  towards  places  which  are 
nearest  the  centre. 

Let  ftey  be  an  arc  of  a  circle  with  centre  a  ; 
the  line  a<5  is  the  shortest  distance  from  cc  to  (iy* 
"  Water  will  run  towards  <3  from  all  Hides  until 
its  surface  becomes  equidistant  from  the  centre. 
It  therefore  follows  that  the  water  takes  up  the 
same  length  on  all  the  lines  radiating  from  the  Fig.  1 

centre.     It  then  remains  in  equilibrium.     But 

the  locus  of  equal  lines  radiating  from  a  centre  in  a  circumference  of  a 
circle.     The  surface  of  the  water,  /5tey,  will  therefore  be  spherical.  " 

Adrastus  (360-317  B.  C),  commenting  on  Aristotle,  made  the  pre- 

*  DUHEM,  0.  5.,  Vol.  II,  p.  II. 


24  THE   ORIGINS 

ceding  proof  precise  in  the  following  terms.  "  Water  will  run  towards 
the  point  <5  until  this  point,  surrounded  by  new  water,  is  as  far  from  oc 
as  /3  and  y.  Similarity,  all  points  on  the  surface  of  the  water  will  be  at 
an  equal  distance  from  a.  Therefore  the  water  exhibits  a  spherical 
form  and  the  whole  mass  of  water  and  Earth  is  spherical.  " 

Adrastus  supplemented  this  proof  with  the  following  evidence, 
which  was  destined  to  become  classical.1 

"  Often,  during  a  voyage,  one  cannot  see  the  Earth  or  an  approach 
ing  ship  from  the  deck,  while  sailors  who  climb  to  the  top  of  a  mast  can 
see  these  things  because  they  are  much  higher  and  thus  overcome  the 
convexity  of  the  sea  which  is  an  obstacle.  " 

We  shall  say  no  more  about  aristotelian  mechanics.  However  inad 
equate  they  may  seem  now,  these  intuitive  theories  have  their  origin  in 
the  most  everyday  observations,  precisely  because  they  take  the  passive 
resistances  to  motion  into  account.  To  an  unsophisticated  observer, 
a  horse  pulling  a  cart  seems  to  behave  according  to  the  law  of  powers, 
in  the  sense  that  it  develops  an  effort  which  increases  regularly  with 
the  speed.  In  order  to  break  away  from  Aristotle's  ideas  and  to  con 
struct  the  now  classical  mechanics,  it  is  necessary  to  disregard  the  va 
rious  ways  in  which  motion  may  be  damped,  and  to  introduce  these 
explicitly  at  a  later  stage  as  factional  forces  and  as  resistances  of  the 
medium.  However  it  may  be,  aristotelian  doctrines  provided  the 
fabric  of  thought  in  mechanics  for  nearly  two  thousand  years,  so  that 
even  Galileo,  who  was  to  become  the  creator  of  modern  dynamics, 
made  his  first  steps  in  science  in  commenting  Aristotle,  and  proved  in 
his  early  writings  to  be  a  faithful  Peripatetic  ;  which,  it  may  be  said  in 
passing,  in  no  way  diminishes  his  glory  as  a  reformer,  on  the  contrary, 
it  only  adds  to  it. 

2.  THE  STATICS  OF  ARCHIMEDES. 

Unlike  Aristotle,  whose  mechanics  is  integrated  into  a  theory  of 
physics  which  goes  so  far  as  to  incorporate  a  system  of  the  world,  Archi 
medes  (287-212  B.C.)  made  of  statics  an  autonomous  theoretical  science, 
based  on  postulates  of  experimental  origin  and  afterwards  supported 
by  mathematically  rigourous  demonstrations,  at  least  in  appearance* 

Here  we  shall  follow  the  treatise  On  the  Equilibrium  of  Planes  or  on 
the  Centres  of  Gravity  of  Planes 2  in  which  Archimedes  discussed  the 
principle  of  the  lever. 

X  ™s  thesis  of  ADRASTUS  is  known  to  us  by  means  of  A  Collection  of  Mathematical 
Knowledge  useful  for  the  Reading  of  Plato,  by  THEON  OF  SMYRNA. 

2  Translation  by  PEYRARD,  Paris,  1807.  The  reader  .should  also  refer  to  that  of 
P.  VER  EECKE,  Paris  and  Anvers,  Desclee  de  Brouwer,  1938. 


HELLENIC    SCIENCE  25 

Archimedes  made  the  following  postulates  as  axioms — 

1)  Equal  weights  suspended  at  equal  distances  (from  a  fulcrum)  are 
in  equilibrium. 

2)  Equal  weights   suspended   at   unequal   distances   cannot  be  in 
equilibrium.     The  lever  will  be  inclined  towards  the  greater  weight. 

3)  If  weights  suspended  at  certain  distances  are  in  equilibrium,  and 
something  is  added  to  one  of  them,  they  will  no  longer  be  in  equilibrium. 
The  lever  will  be  inclined  towards  the  weight  which  has  been  increased. 

4)  Similarily,  if  something  is  taken  away  from  one  of  the  weights, 
they  will  no  longer  be  in  equilibrium,  but  will  be  inclined  towards  the 
weight  which  has  not  been  decreased. 

5)  If  equal  and  similar  plane  figures  coincide,  their  centres  of  gravity 
will  also  coincide. 

(The  concept  of  Centre  of  Gravity  appears  to  have  been  defined 
by  Archimedes  in  an  earlier  manuscript,  of  which  no  trace  remains.) 

6)  The  centres  of  gravity  of  unequal  but  similar  figures  are  similarity 
placed. 

7)  If  magnitudes  suspended  at  certain  distances  arc  in.  equilibrium, 
equivalent  magnitudes  suspended  at  the  same  distances  will  also  be  in 
equilibrium. 

8)  The  centre  of  gravity  of  a  figure  which  is  nowhere  concave  is 
necessarily  inside  the  figure. 

With  this  foundation,  Archimedes  proved  the  following  propositions. 

Proposition  I.  —  When  weights  suspended  at  equal  distances  are  in 
equilibrium,  these  weights  are  equal  to  each  other. 

(Proof  by  reductio  ad  absurdum  based  on  Postulate  4).) 

Proposition  II.  —  Unequal  weights  suspended  at  equal  distances 
will  not  be  in  equilibrium,  bxit  the  greater  weight  will  fall. 
(Proof  based  on  Postulates  1)  and  3).) 

Proposition  III.  —  Unequal  weights  suspended  at  unequal  distances 
may  be  in  equilibrium,  in  which  case  the  greater  weight  will  be  suspended 
at  the  shorter  distance. 

(Proof  based  on  Postulates  4),  1)  and  2).  Thin  proof  only  confirms 
the  second  part  of  the  proposition,  and  does  not  demonstrate  the  possi- 
bility  of  the  equilibrium  of  two  unequal  weights.  This  must  be  regarded 
as  an  additional  postulate  of  experimental  origin.) 


26 


THE    ORIGINS 


Proposition  IV.  —  If  two  equal  magnitudes  do  not  have  the  same 
centre  of  gravity,  the  centre  of  gravity  of  the  magnitude  made  up  of 
these  two  magnitudes  is  the  point  situated  at  the  middle  of  the  line 
which  joins  their  centres  of  gravity. 

(The  proof,  based  on  Postulate  2),  is  a  demonstration  by  reductio  ad 
absurdum  which,  moreover,  assumes  that  the  centre  of  gravity  of  the 
combined  magnitude  lies  on  the  line  joining  the  centres  of  gravity  of 
the  component  magnitudes.) 

Proposition  V.  —  If  the  centres  of  gravity  of  three  magnitudes  lie 
on  the  same  straight  line,  and  if  the  magnitudes  are  equally  heavy  and 
the  distances  between  their  centres  of  gravity  are  equal,  the  centre  of 
gravity  of  the  combined  magnitude  will  be  the  point  which  is  the  centre 
of  gravity  of  the  central  magnitude. 

(This  is  a  corollary  of  Proposition  IV,  which  Archimedes  later 
extended  to  the  case  of  n  magnitudes.  The  enunciation  is  suitably 
modified  if  n  is  even.) 

Proposition  VI.  —  Commensurable  magnitudes  are  in  equilibrium 
when  they  are  reciprocally  proportional  to  the  distances  at  which  they 
are  suspended. 


A 

2 

8 

L                           E             C            H            D            K 

f                                                          1                             1                             1                             1                             1 

Fig.  2 

"  Let  the  commensurable  magnitudes  be  A  and  B,  and  let  their 
centres  of  gravity  be  the  points  A  and  B.  Let  ED  be  a  certain  length 
and  suppose  that  the  magnitude  A  is  to  the  magnitude  B  as  the  length 
DC  is  to  the  length  CE.  It  is  necessary  to  prove  that  the  centre  of 
gravity  of  the  magnitude  formed  of  the  two  magnitudes  A  and  B  i« 
the  point  C. 

"  Since  A  is  to  B  as  DC  is  to  CE  and  since  the  areas  A  and  B  are 
commensurable,  the  lengths  DC  and  CE  will  also  be  commensurable. 
Therefore  the  lengths  EC  and  CD  have  a  common  measure,  say  IV. 


HELLENIC   SCIENCE  27 

Suppose  that  each  of  the  lengths  DH  and  DK  is  equal  to  the  length  EC, 
and  that  the  length  EL  is  equal  to  the  length  DC.  Since  the  length  DH 
is  equal  to  the  length  CJ5,  the  length  DC  will  be  equal  to  the  length  .EH, 
and  the  length  LE  will  be  equal  to  the  length  EH.  Hence  the  length 
LH  is  twice  the  length  DC,  and  the  length  HK  twice  the  length  CE. 
Therefore  the  length  N  is  a  common  measure  of  the  lengths  DH  and  HK 
since  it  is  a  common  measure  of  their  halves.  But  A  is  to  B  as  DC  is 
to  CE,  so  that  A  is  to  B  as  LH  is  to  HK.  Let  A  be  as  many  times 
greater  than  Z  as  LH  is  greater  than  N.  The  length  LH  will  be  to  the 
length  IV  as  A  is  to  Z.  But  KH  is  to  LH  as  B  is  to  A.  Therefore,  by 
equality,  the  length  KH  is  to  the  length  N  as  B  is  to  Z.  Then  B  is  as 
many  times  greater  than  Z  as  KH  is  a  multiple  of  N.  But  it  has  been 
arranged  that  A  is  also  a  multiple  of  Z,  Therefore  Z  is  a  common 
measure  of  A  and  B.  Consequently,  if  LH  is  divided  into  segments 
each  equal  to  JV,  and  A  into  segments  each  equal  to  Z,  A  will  contain 
as  many  segments  equal  to  Z  as  LH  contains  segments  equal  to  N. 
Therefore,  if  a  magnitude  equal  to  Z  is  applied  to  each  segment  of  LH 
in  such  a  way  that  its  centre  of  gravity  is  at  the  centre  of  the  segment, 
all  the  magnitudes1  will  be  equal  to  A.  Further,  the  centre  of  gravity 
of  the  magnitude  made  up  of  all  these  magnitudes  will  be  the  point  JS, 
remembering  that  they  are  an  even  number  and  that  LE  is  equal  to  HE 
(Proposition  V).  Similarity  it  could  be  shown  that  if  a  magnitude 
equal  to  Z  was  applied  to  each  of  the  segments  of  KH>  with  its  centre 
of  gravity  at  the  centre  of  each  segment,  all  those  magnitudes  l  would 
be  equal  to  B  and  that  the  combined  centre  of  gravity  would  be  D. 
But  the  magnitude  A  is  applied  at  the  point  E  and  the  magnitude  B  at 
the  point  D.  Therefore  certain  equal  magnitudes  are  placed  on  the 
same  line,  their  centres  of  gravity  are  separated  from  each  other  by  the 
same  interval  and  they  are  an  even  number.  It  is  therefore  clear  that 
the  centre  of  gravity  of  the  magnitude  composed  of  all  these  magnitudes 
is  the  point  at  the  middle  of  the  line  on  which  the  centres  of  gravity  of 
the  central  magnitudes  lie  (Proposition  V).  But  the  length  LJE  is 
equal  to  the  length  CD  and  the  length  EC  to  the  length  CK.  Thus  the 
centre  of  gravity  of  the  magnitude  composed  of  all  these  areas  is  the 
point  C,  Therefore,  if  the  magnitude  A  is  applied  to  the  point  E  and 
the  magnitude  B  to  the  point  D,  the  two  areas  will  be  in  equilibrium 
about  the  point  C.  " 

Archimedes  then  extended  this  proportion  to  the  caae  of  magnitudes 
A  and  B  which  were  incommensurable.  This  demonstration  depended 
on  the  method  of  exhaustion.  We  have  reproduced  this  proof  of 

1  Read,  **  the  combination  of  ali  these  magnitudes.  " 


28  THE    ORIGINS 

Proposition  VI  in  its  entirety  in  order  to  illustrate  the  nature  of  Archi 
medes'  logical  apparatus.  This  should  not  be  allowed,  however,  to 
create  too  great  an  illusion  of  power. 

Indeed,  Archimedes  assumes  in  this  proof  that  the  load  on  the 
fulcrum  of  a  lever  is  equal  to  the  sum  of  the  two  weights  which  it 
supports.1  Further,  he  made  use  of  the  principle  of  superposition 
of  equilibrium  states,  without  emphasising  that  this  was  an  experi 
mental  postulate.  Finally,  and  this  is  the  most  telling  objection  to 
the  proceeding  analysis,  Archimedes,  together  with  those  of  his  suc 
cessors  who  tried  to  improve  his  proof,  tacitly  made  the  hypothesis 
that  the  product  PL  measures  the  effect  of  a  weight  P  placed  at  a 
distance  L  from  a  horizontal  axis.  In  fact,  in  the  case  of  complete 
symmetry  which  is  envisaged  in  Archimedes'  first  postulate,  equili 
brium  obtains  whatever  law  of  the  form  Pf(L)is  taken  as  a  measure 
of  the  effect  of  the  weight  P.  It  is  therefore  impossible,  with  the  help 
of  Archimedes'  postulates  alone,  to  substantiate  Proposition  VI  in  a 
logical  way.2 

For  the  rest,  the  treatise  On  the  Equilibrium  of  Planes  is  concerned 
with  the  determination  of  the  centres  of  gravity  of  particular  geome 
trical  figures.  After  having  obtained  the  centres  of  gravity  of  a  tri 
angle,  a  parallelogram  and  a  trapezium,  Archimedes  determined  the 
centre  of  gravity  of  a  segment  of  a  parabola  by  means  of  an  analysis 
which  is  a  milestone  in  the  history  of  mathematics  (Book  II,  Pro 
position  VIII). 

We  shall  now  concern  ourselves  with  Archimedes'  treatise  on 
Floating  Bodies.  The  author  starts  from  the  following  hypothesis— 

66  The  nature  of  a  fluid  is  such  that  if  its  parts  are  equivalently 
placed  and  continuous  with  each  other,  that  which  is  the  least  compress 
ed  is  driven  along  by  that  which  is  the  more  compressed.  Each  part 
of  the  fluid  is  compressed  by  the  fluid  which  is  above  it  in  a  vertical 
direction,  whether  the  fluid  is  falling  somewhere  or  whether  it  in 
driven  from  one  place  to  another.  " 

From  this  starting  point,  the  following  propositions  derive  in  a 
logical  sequence. 

Proposition  I.  —  If  a  surface  is  intersected  by  a  plane  which  always* 
passes  through  the  same  point  and  if  the  section  is  a  circumference 
(of  a  circle)  having  this  fixed  point  as  its  centre,  the  surface  in  that 
of  a  sphere. 

1  This  is  a  point  which  can  be  established  rigorously  by  considerations  of  symmetry 
alone,  as  FOURIER  was  to  show,  much  later,  in  his  perfection  of  a  similar  attempt  due 
to  D'ALEMBERT. 

2  Cf.  MACH,  Mechanics,  p.  21.     Throughout  this  work,  Mach's  treatise  will  be 
indicated  by  the  letter  M. 


HELLENIC    SCIENCE 


29 


Proposition  II.  —  The  surface  of  any  fluid  at  rest  is  spherical  and 
the  centre  of  this  surface  is  the  same  as  the  centre  of  the  Earth. 
This  result  had  already,  as  we  have  seen,  been  enunciated  by  Aristotle. 

Proposition  III.  —  If  a  body  whose  weight  is  equal  to  that  of 
the  same  volume  of  a  fluid  (a)  is  immersed  in  that  fluid,  it  will  sink 
until  no  part  of  it  remains  above  the  surface,  but  will  not  descend 
further. 

We  shall  reproduce  the  proof  of  this  proposition,  which  is  the 
origin  of  Archimedes'  Principle. 

"  Let  a  body  have  the  same  heaviness  as  a  liquid.  If  this  is  possi 
ble,  suppose  that  the  body  is  placed  in  the  fluid  with  part  of  it  above 
the  surface.  Let  the  fluid  be  at  rest.  Suppose  that  a  plane  which 
passes  through  the  centre  of  the  Earth  intersects  the  fluid  and  the 


f. 


body  immersed  in  it  in  such  a  way  that  the  section  of  the  fluid  is  Alt(!D 
and  the  section  of  the  body  is  ERTF.  Let  K  be  the  centre  of  the 
Earth,  BHTC  be  the  part  of  the  body  which  is  immersed  in  the  fluid 
and  DEFC  the  part  which  projects  out  of  it.  Construct  a  pyramid 
whose  base  is  a  parallelogram  in  the  surface  of  the  fluid  (a)  and  whose 
apex  is  the  centre  of  the  Earth.  Let  the  intersection  of  the  faces 
of  the  pyramid  by  the  plane  containing  the  are  ABCD  be  KL  and 
KM.  In  the  fluid,  and  below  EFTII  draw  another  spherical  surface 
XOP  having  the  point  K  as  its  centre,  in  such  a  way  that  XOP  is  the 
section  of  the  surface  by  the  plane  containing  the  are  ABCD.  Take 
another  pyramid  equal  to  the  first,  with  which  it  is  contiguous  and 
continuous,  and  such  that  the  sections  of  its  faces  are  KM  and  KN. 
Suppose  that  there  is,  in  the  fluid,  another  solid  RSQY  which  is  made 
of  the  fluid  and  is  equal  and  similar  to  BHTC,  that  part  of  the  body 


30  THE   ORIGINS 

EHTF  which,  is  immersed  in  the  fluid.     The  portions  of  the  fluid 
which  are  contained  by  the  surface  XO  in  the  first  pyramid  and  the 
surface  OP  in  the  second  pyramid  are  equally  placed  and  continuous 
with  each  other.     But  they    are  not    equally    compressed.     For    the 
portions  of  the  fluid  contained  in  XO  are  compressed  by  the  body 
EHTF  and  also  by  the  fluid  contained  by  the  surfaces  LM,  XO  and 
those  of  the  pyramid.     The  parts   contained  in  PO  are  compressed 
by  the  solid  RSQY  and  by  the  fluid  contained  by  the  surfaces  OP,  MN 
and  those  of  the  pyramid.     But  the  weight  of  the   fluid   contained 
between  MN  and  OP  is  less  than  the  combined  heaviness  of  the  fluid 
between  LM  and  XO  and  the  solid.     For  the  solid  RSQY  is  smaller 
than  the  solid  EHTF— RSQY  is  equal  to  BHTC—  and  it  has  been 
assumed  that  the  body  immersed  has,  volume  for  volume,  the  same 
heaviness  as  the  fluid.     If  therefore  one  takes  away  the  parts  which 
are  equal  to  each  other,  the  remainder  will  be  unequal.     Consequently 
it  is  clear  that  the  part  of  the  fluid  contained  in  the  surface  OP  will 
be  driven  along  by  the  part  of  the  fluid  contained  in  the  surface  JtO, 
and  that  the  fluid  will  not  remain  at  rest.     Therefore,  no  part  of  the 
body  which  has  been  immersed  will  remain  above  the  surface.     How 
ever,  the  body   will   not  fall   further.     For  the  body  has  the    same 
heaviness  as  the  fluid  and  the  equivalently  placed  parts  of  the  fluid 
compress  it  similarily.  " 

Proposition  IV.  —  If  a  body  which  is  lighter  than  a  fluid  is  placed 
in  this  fluid,  a  part  of  the  body  will  remain  above  the  surface. 
(Proof  analogous  to  that  of  Proposition  III,) 

Proposition  V.  —  If  a  body  which  is  lighter  than  a  fluid  is  placed 
in  the  fluid,  it  will  be  immersed  to  such  an  extent  that  a  volume  of 
fluid  which  is  equal  to  the  volume  of  the  part  of  the  body  immersed 
has  the  same  weight  as  the  whole  body. 

The  diagram  is  the  same  as  the  preceding  one  (Proposition  III). 
"  Let  the  liquid  be  at  rest  and  the  body  EHTF  be  lighter  than 
the  fluid.  If  the  fluid  is  at  rest,  parts  which  are  equivalently  placed 
will  be  similarly  compressed.  Then  the  fluid  contained  by  each  of 
the  surfaces  XO  and  OP  is  compressed  by  an  equal  weight.  But, 
if  the  body  BHTC  is  excluded,  the  weight  of  fluid  in  the  first  pyramid 
is  equal,  with  the  exclusion  of  the  fluid  RSQY,  to  the  weight  of  fluid 
in  the  second  pyramid.  Therefore  it  is  clear  that  the  weight  of  the 
body  EHTF  is  equal  to  the  weight  of  the  fluid  RSQ  Y.  From  which 
it  follows  that  a  volume  of  fluid  equal  to  that  of  the  body  which  is 
immersed  has  the  same  weight  as  the  whole  body.  " 


HELLENIC   SCIENCE 


31 


Proposition  VI.  —  If  a  body  which  is  lighter  than  a  fluid  is  totally 
and  forcibly  immersed  in  it,  the  body  will  be  thrust  upwards  with 
a  force  equal  to  the  difference  between  its  weight  and  that  of  an  equal 
volume  of  fluid. 

Proposition  VII.  —  If  a  body  is  placed  in  a  fluid  which  is  lighter 
than  itself,  it  wiU  fall  to  the  bottom.  In  the  fluid  the  body  will  be 
lighter  by  an  amount  which  is  the  weight  of  the  fluid  which  has  the 
same  volume  as  the  body  itself. 

The  first  Book  of  the  treatise  On  Floating  Bodies  concludes  with 
the  following  hypothesis—  "  We  suppose  that  bodies  which  are  thrust 
upwards  aU  follow  the  vertical  which  passes  through  their  centre  of 

gravity. "  .  . 

In  Book  II,  Archimedes  modified  the  principle  which  is  the  subject 

of  Proposition  V,  Book  I,  to  the  following  form— 

"  If  any  solid  magnitude  which  is  lighter  than  a  fluid  is  immersed 
in  it,  the  proportion  of  the  weight  of  the  solid  to  the  weight  of  an 
equal  volume  of  fluid  will  be  the  same  as  the  proportion  of  the  volume 
of  that  part  of  the  solid  which  is  submerged  to  the  volume  of  the  whole 

solid-"  -       i.  i.  A     i.-      A 

We  shall  pass  over  the  proof  of  this  proposition,  in  which  Archimedes 

once  more  deploys  that  powerful  logical  apparatus  with  which  we  are 
now  familiar.  The  rest  of  Book  II  is  devoted  to  a  detailed  study  of 
the  equilibrium  of  a  floating  body  which  is  shaped  like  a  right  segment 
of  a  "  parabolic  conoid. "  In  Archimedes'  language  (in  the  treatise 
On  Conoids  and  Spheroids),  this  term  refers  to  the  figure  which  we 
would  now  call  a  parabolic  cylinder.  It  may  be  surmised  that  Archi 
medes  was  interested  in  this  problem  for  a  most  practical  reason,  for 
this  surface  is  similar  to  that  of  the  hull  of  a  ship. 

It  is  of  interest  that,  throughout  this  study,  Archimedes  approxi 
mated  the  free  surface  of  a  fluid  by  a  plane,  and  that  he  treated  verticals 
an  if  they  were  parallel.  This  is  necessary  if  the  concept  of  centre 
of  gravity  is  to  be  utilised.  Thus  Archimedes  must  have  understood 
the  necessity  and  the  practical  importance  of  this  approximation, 
even  though  his  principle  was  based  on  the  convergence  of  the  verticals 
at  the  centre  of  the  Earth,  the  spherical  symmetry  of  fluid  surfaces 
and  a  rather  vague  hypothesis  about  the  pressures  obtaining  in  the 
interior  of  a  fluid. 


CHAPTER   TWO 


ALEXANDRIAN  SOURCES  AND  ARABIC  MANUSCRIPTS 


1.  THE  "  MECHANICS  "  OF  HERO  OF  ALEXANDRIA. 

It  seems  that  Hero  of  Alexandria  lived  at  some  time  during  the 
Ilnd  Century  A.  D.  His  treatise  Mechanics  discusses  certain  simple 
machines — the  lever,  pulley-block  and  the  screw— alone  or  in  various 
combinations,  and  is  only  available  to  us  in  the  form  of  an  arabic  version 
which  has  been  translated  and  published  by  Carra  de  Vaux.1 

As  far  as  it  concerns  the  history  of  mechanics,  the  essential  import 
ance  of  this  work  lies  in  the  fact  that  its  author  used  the  now  classical 
idea  of  moment  in  his  discussion  of  a  lever  which  was  not  straight. 
Whether  or  not  this  conception  was  an  original  one  remains  doubtful. 
Indeed  alexandrian  learning  had  access  to  a  treatise  of  Archimedes  that 
was  devoted  to  levers  (TZegJ  £vy&v)  and  in  which  the  problem  of  the 

angular  lever  had  been  treated.     No  trace 
of  this  writing  is  extant. 

However  this  may  be,  we  shall  quote 
from  Hero's  Mechanics. 

"  Consider  an  arm  of  a  balance  which 
does  not  have  the  same  thickness  or 
heaviness  throughout  its  length.  It  may 
be  made  of  any  material.  It  is  in  equili 
brium  when  suspended  from  the  point  }' 
— by  equilibrium  we  understand  the  arrant 
of  the  beam  in  a  stable  position,  even 
though  it  may  be  inclined  in  one  direction 
or  the  other.  Now  let  weights  be  suspended 
at  some  points  of  the  beam— say  at  b 

and  £.     The  beam  will  take  up  a  new  position  of  equilibrium  after  the 
weights  have  been   hung  on.     Archimedes   has  shown,  in  this  case  as 


1  Journal  asiatique,  1891. 


ALEXANDRIAN    SOURCES    AND    ARABIC    MANUSCRIPTS 


33 


well,  that  the  relation  of  the  weights  to  each  other  is  the  same  as  the 
inverse  relation  of  the  respective  distances. 1  Concerning  these  distances 
in  the  case  of  irregular  and  inclined  beams,  it  should  be  imagined  that 
a  string  is  allowed  to  fall  from  y  towards  the  point  £•  Construct  a 
line  which  passes  through  the  point  £ — the  line  77  £0 — and  which  should 
be  arranged  to  intersect  the  string  at  right  angles.  When  the  beam  is 
at  rest  the  relation  of  £77  to  £0  is  the  same  as  the  relation  of  the  weight 
hung  at  the  point  e  to  the  weight  hung  at  the  point  <5.  " 

Hero  employed  a  similar  argument  in  his  discussion  of  the  wheel  and 
axle.  In  fact,  in  reducing  the  study  of  these  machines  to  the  principle 
of  circles  he  was  implicitly  using  the  notion  of  moment.  Thus  it  is 
clear,  though  not  explicitly  stated,  that  in  his  discussion  of  the  axle 
Hero  understands  that  a  weight  £  can  be  replaced  by  an  equal  force 
applied  tangentially  at  A,  because  AF  has  the  same  moment  as  £.2 


2.    PAPPUS'S    THEORIES    OF    THK    INCLINED    PLANE    AND    OF    THE    CENTRE 
OF  GRAVITY. 

Pappus  (IVth  Century  A.  D.)  appears  to  be  the  only  geometer  of 
Antiquity  who  took  up  the  problem  of  the  motion  arid  equilibrium  of 
a  heavy  body  on  an  inclined  plane.  The  proof  that  we  shall  analyse 
now  is  taken  from  Book  VIII  of  his  (Collections  (From  among  the  varied 
and  delightful  problems  of  mechanics)  . 

Pappus  assumes  that  a  certain  effort  y  is  necessary  to  move  a  weight 
a  on  the  horizontal  plane  fiv,  and  that  a  power  0  is  necessary  to  draw  it 


1  OAKKA  I>E  YAVX'S  nurmiHe  that  Hero  in  referring  to  the  treatise 
probably  correct. 

2  Cf.  JorcuET,  Lectures  de  Mvcaniquv,  Vol.  I,  p.  215.     Throughout  the  present 
book  thin  treatise  will  be  indicated  by  the  letterH  L.  M. 


34  THE   ORIGINS 

up  the  inclined  plane  px.     He  sets  out  to  determine  the  relation  between 

y  and  0. 

The  weight  a  on  the  plane  px  has  the  form  of  a  sphere  with  centre  e. 
Pappus  reduces  the  investigation  of  the  equilibrium  of  this  sphere  on 


Fig.  6 


the  inclined  plane  to  the  following  problem.  A  balance  supported  at  A 
carries  the  weight  oc  at  &  and  the  weight  ft  which  is  necessary  to  keep 
it  in  equilibrium  at  y  —  the  end  of  the  horizontal  radius  eq.  The  law  of 
the  angular  lever,  which  Pappus  borrows  from  Archimedes'  UeQl 
or  from  Hero's  Mechanics,  provides  the  relation 


On  the  horizontal  plane  where  the  power  necessary  to  move  a  is  y, 
the  power  necessary  to  move  along  /?  will  be 


Pappus  then  assumes  that  the  power  0  that  is  able  to  move  the 
weight  oc  on  the  inclined  plane  px  will  be  the  sum  of  the  powers  <3  and 
y,  that  is 


ALEXANDRIAN    SOURCES    AND    ARABIC    MANUSCRIPTS  35 

Evidently  the  necessity  of  a  power  y  for  pulling  the  weight  a  on  the 
horizontal  plane  derives  from  Aristotle's  dynamics,  in  which  all  unnat 
ural  motion  requires  a  motive  agency.  The  argument  by  which  Pappus 
introduces  the  lever  eArj  supporting  the  two  weights  a  and  ft  is  rather  a 
natural  one,  even  though  it  does  not  lead  to  a  correct  evaluation  of  ft. 
The  last  hypothesis,  concerning  the  addition  of  d  and  y,  the  powers 
that  are  necessary  to  move  ft  and  a  respectively  on  the  horizontal  plane 
is,  on  the  other  hand,  most  strange. 

However  incorrect  it  may  have  been,  this  proof  was  destined  to 
inspire  the  geometers  of  the  Renaissance.  Guido  Ubaldo  was  to  adopt 
it  and  Galileo  was  to  be  occupied  in  demonstrating  its  falsehood. 

Archimedes  certainly  formulated  a  precise  definition  of  centre  of 
gravity,  but  there  is  no  trace  of  anything  of  this  kind  in  those  of  his 
writings  that  are  available  to  us.  Therefore  it  is  of  some  value  to 
record  the  definition  which  is  due  to  Pappus. 

Imagine  that  a  heavy  body  is  suspended 
by  an  axis  a/?  and  let  it  take  up  its  equili 
brium  position.  The  vertical  plane  pass 
ing  through  oc/J  u  will  cut  the  body  into 
two  parts  that  are  in  equilibrium  with  each 
other  and  which  will  be  hung  in  such  a  way, 
on  one  side  of  the  plane  and  on  the  other, 
as  to  be  equal  with  respect  to  their  weight.  " 
Taking  another  axis  &.'ftf  and  repeating  the 
same  operation,  the  second  vertical  plane  n^" 

passing  through  a'/?'  will  certainly  cut  the 

first — if  it  were  parallel  to  it  "  each  of  these  two  planes  would  divide 
the  body  into  two  equal  parts  which  would  be  at  the  same  time  of  equal 
weight  and  of  unequal  weight,  which  is  absurd.  " 

Now  suspend  the  body  from  a  point  y  and  draw  the  vertical  yd 
through  the  point  of  suspension  when  equilibrium  is  established.  Take 
a  second  point  of  suspension  yr  and,  in  the  same  way,  draw  the  vertical 
y'd'.  The  two  lines  yd,  y'd'  necessarily  intersect.  For  if  not,  through 
each  of  them  a  plane  could  be  drawn  so  as  to  divide  the  body  into  two 
parts  in  equilibrium  with  each  other,  and  in  such  a  way  that  these  two 
planes  were  parallel  to  each  other.  This  is  impossible. 

All  lines  like  yd  will  therefore  intersect  at  one  unique  point  of  the 
body  that  is  called  the  centre  of  gravity. 

Pappus  did  not  distinguish  clearly,  as  Guido  Ubaldo  was  to  do  in  his 
Commentary  on  Archimedes*  two  books  on  the  equilibrium  of  weights  (1588) 
between  "equiponderant"  parts,  that  is  parts  that  are  in  equilibrium,  in 
the  positions  which  they  occupy,  and  parts  which  have  the  same  weight. 


36  THE    ORIGINS 

3.    THE  FRAGMENTS  ATTRIBUTED  TO  EUCLID  IN  ARABIC  WRITINGS. 

Greek  antiquity  does  not  attribute  any  work  on  mechanics  to  Euclid. 
However  his  name  occurs  frequently  in  this  connection  in  the  writings 
of  arahic  authors. 

Euclid's  book  on  the  balance,  an  arabic  manuscript  of  970  A.  D. 
which  has  been  brought  to  light  by  Dr.  Woepke,1  seems  to  have  remain 
ed  unknown  to  the  western  Middle  Ages.  This  relic  of  greek  science  may 
be  contemporaneous  with  Euclid  and  may  thus  antedate  Archimedes. 
It  contains  a  geometrical  proof  of  the  law  of  levers  which  is  independent 
of  Aristotle's  dynamics  and  which  makes  explicit  appeal  to  the  hypo 
thesis  that  the  effect  of  a  weight  P  placed  at  the  end  of  an  arm  of  a  lever 
is  expressed  by  the  product  PL.  We  have  had  occasion  to  emphasise 
the  necessity  of  this  hypothesis  in  our  analysis  of  Archimedes'  proof. 

The  treatise  Liber  Euclidis  de  gravi  et  levi,  often  simply  called  De 
ponderoso  et  Zevi,  has  been  known  for  a  long  time.  It  includes  a  very 
precise  exposition  of  aristotelian  dynamics  which  is  arranged,  after 
Euclid's  style,  in  the  form  of  definitions  and  propositions.  The  latin 
text  renders  the  terms  dvvKjLus  and  ioyvQ,  by  which  Aristotle  meant 
"  power  ",  as  virtus  and  fortitude.  Bodies  that  travel  equal  distances 
in  the  same  medium — air  or  water — in  times  which  are  equal  to  each 
other,  are  said  to  be  equal  in  virtus.  Bodies  that  travel  equal  distances 
in  unequal  times  are  of  different  virtus,  and  that  which  takes  the  shorter 
time  is  said  to  have  the  greater  virtus.  Bodies  of  the  same  kind  are 
those  that,  volume  for  volume,  are  equal  in  virtus.  That  which  lias 
the  greater  virtus  is  said  to  be  solidius  (more  dense). 

The  virtus  of  bodies  of  the  same  kind  is  proportional  to  their  dimen 
sions  ;  that  is,  the  bodies  fall  with  velocities  which  are  proportional  to 
their  volume.  If  two  heavy  bodies  are  joined  together,  the  velocity 
with  which  the  combination  will  fall  will  be  the  sum  of  the  velocities 
of  the  separate  bodies. 

Duhem  has  found,  in  a  XlVth  Century  manuscript,2  four  proposi 
tions  on  questions  in  statics  which  complete  De  ponderoso  et  kvi.  This 
manuscript  contains  a  theory  of  the  roman  balance,  and  shows  that  the 
fact  that  the  balance  is  a  heavy  homogenous  cylinder  does  not  alter  the 
relation  of  the  weights  to  each  other. 

Finally,  in  a  Xlllth  Century  manuscript,  Duhem  has  unearthed  a 
text  called  Liber  Euclidis  de  ponderibus  secundum  terminorum  circonfe- 
rentiam  3  which  connects  the  law  of  levers  with  aristotelian  dynamics 
and  also  contains  a  theory  of  the  roman  balance. 

1  Journal  asiatique,  Vol.  18,  1851,  p.  217. 

2  BMiothSque  Nationale,  Paris,  latin  collection,  Ms.  10,260 

3  Ibid.,  Ms.  16,649. 


ALEXANDRIAN    SOURCES    AND    ARABIC   MANUSCRIPTS 


37 


4.  THE  BOOK  OF  CHARISTION. 


Liber  Charastonis  is  the  latin  version  of  an  arabic  text  due  to 
the  geometer  Thabit  ibn  Kurrah  (836-901).  The  original  greek  version 
remains  unknown,  and  the  question  of  whether  karaston  (in  Arabic — 
karstun)  refers  merely  to  the  roman  balance  or  to  the  name  of  the  greek 
geometer  Charistion  (a  contemporary  of  Philon  of  Byzantium  in  the 
Ilnd  Century  B.  C.)  has  been  the  subject  of  much  scholarly  debate. 

We  shall  follow  Duhem l  in  summarising  the  theory  of  the  roman 
balance  which  is  found  in  Liber  Charastonis, 


<f          9 


Fig.  8 


A  heavy  homogeneous  cylindrical  beam  ab  whose  arms  ag  and  bg 
are  unequal  may  be  maintained  in  a  horizontal  position  by  means  of  a 
weight  e  hung  from  the  end  of  the  shorter  arm  ag.  If  bd  is  the  amount 
by  which  the  longer  arm  exceeds  the  shorter  arm  and  u  is  the  centre  of 
&<J,  the  weight  e  will  be  to  bd  as  gu>  is  to  ga.  If  p  is  the  total  weight  of 
the  beam 

db 


If  this  weight  were  known  it  could  be  represented  exactly  by  a 
scale-pan  hung  from  the  shorter  arm,  and  the  karaston  arranged  in  this 
way  could  be  treated  as  a  weightless  beam. 

We  must  also  mention,  as  one  of  the  sources  of  statics,  the  treatise 
De  Canoniof  a  latin  translation  of  a  greek  text  which  adds  nothing 
essential  to  Liber  Charastonis. 


1  0.  S.,  Vol.  T,  p.  90. 

2  Bibliothvqiie  Nationale^  Paris.,  latin  collection,  Ms.  737B  A. 


CHAPTER   THREE 

THE  XHIth  CENTURY 
THE  SCHOOL  OF  JORDANUS 


1.  JORDANUS  OF  NEMORE  AND  "  GRAVITAS  SECUNDUM  SITUM.  " 

The  Middle  Ages  had  access  to  the  Problems  of  Mechanics  and  to  the 
works  of  Aristotle.  They  had  also  inherited  the  fragments  attributed 
to  Euclid — with  the  exception  of  the  Book  on  the  Balance — as  well  as 
the  Liber  Charastonis  from  arabic  learning.  They  had  no  knowledge 
of  Archimedes,  Hero  of  Alexandria  and  Pappus. 

In  spite  of  the  researches  of  the  scholars,  the  personality  of  Jordanus 
remains  mysterious.  At  least  three  XHIth  Century  manuscripts  on 
statics  have  been  attributed  to  him,  although  these  are  clearly  in  the 
style  of  different  authors.  Neither  Jordanus's  nationality  nor  the  pe 
riod  in  which  he  lived  is  known  with  any  certainty.  Daunou  believes 
him  to  have  lived  in  Germany  about  1050,  Chasles  associates  him  with 
the  Xlllth  Century  while  Curtze  places  him  about  1220  under  the  name 
of  Jordanus  Saxo.  Michaud  has  identified  him  with  Raimond  Jordan, 
provost  of  the  church  of  Uz£s  in  1381  which  is  clearly  too  late.  With 
Montucla,  we  shall  here  adopt  the  intermediate  view  that  associates 
Jordanus  of  Nemore  with  the  Xlllth  Century. 

Like  Duhem,  we  shall  follow  the  Elementa  Jordani  super  demonstra* 
tionem  ponderis.1  This  work  comprised  seven  axioms  or  definitions 
followed  by  nine  propositions.  The  essential  originality  of  Jordanus  lay 
in  the  systematic  use,  in  his  study  of  the  motion  of  heavy  bodies,  of  the 
effective  path  in  a  vertical  direction  as  a  measure  of  the  effect  of  a  weight, 
which  was  usually  placed  at  the  end  of  a  lever  and  described  a  circle  m 
consequence.  Thus  his  statics  stems,  implicitly,  from  the  principle  of 
virtual  work.  The  word  work,  taken  in  the  modern  sense,  is  to  be  con- 

1  Bibliothdque  Nationals,  Paris,  Ms.  10,252,  dated  1464.  There  also  exists  an  in 
complete  manuscript  of  the  same  work,  dating  from  the  Xlllth  Century,  in  the  Biblio- 
theque  Mazarine,  Ms.  3642. 


THE   XlHth   CENTURY  39 

trasted  with  the  word  velocity  and  with  the  concept  of  virtual  velocities 
which  may  be  traced  in  the  arguments  of  Problems  of  Mechanics.  Of 
course  Jordanus  never  used  the  word  "  work  "  itself.  He  considered 
the  heaviness  of  a  particle  relative  to  its  situation  (gravitas  secundum 
situm)  without  making  clear  the  relation  that  exists  between  this  quan 
tity  and  the  heaviness  in  the  strict  sense. 

Jordanus  formulated  his  principle  in  a  picturesque  Latin  which  merits 
quotation. 

"  Omnis  ponderosi  motum  esse  ad  medium,  virtutemque  ipsius  poten- 
tiam  ad  inferiora  tendendi  et  motui  contrario  resistendi. 

"  Gravius  esse  in  descendendo  quando  ejusdem  motus  ad  medium  rectior. 

"  Secundum  situm  gravius,  quando  in  eodem  situ  minus  obliquus  est 
descensus. 

64  Obliquiorem  autem  descensum  in  eadem  quantitate  minus  capere  de 
directo.  " 

Or— 

"  The  motion  of  all  heavy  things  is  towards  the  centre,1  its  strength 
being  the  power  which  it  has  of  tending  downwards  and  of  resisting  a 
contrary  motion. 

"  A  moving  body  is  the  heavier  in  its  descent  as  its  motion  towards 
the  centre  is  the  more  direct. 

46  A  body  is  the  heavier  because  of  its  situation  as,  in  that  situation, 
its  descent  is  the  less  oblique. 

44  A  more  oblique  descent  is  one  that,  for  the  same  path,  takes  less 
of  the  direct.  " 

Thus  a  certain  weight  placed  at  6, 
at  the  end  of  the  lever  c6,  has  a  smaller 
gravity  secundum  situm  than  the  same 
weight  has  when  it  is  at  a,  at  the  end 
of  the  horizontal  radius  ca.  Indeed,  on 
the  circumference  of  the  circle  with 
centre  c  and  radius  ca  =--  c6,  if  the 
body  falls  from  b  to  h  along  the  arc 
oh  the  effective  path  in  a  vertical 
direction  is  b'  hf.  On,  the  other  hand 
if  the  body  starts  from  a  and  falls 
along  an^arc  c5,  which  is  equal  to 
the  arc  bh>  the  effective  vertical  path 

is  czr  and  is  greater  than  V  h*.  Thus  the  descent  6A,  equal  to  the 
descent  oz,  is  more  oblique  than  that  and  takes  less  of  the  direct. 

1  Understood  as  the  common  centre  of  all  heavy  things  in  Aristotle's  sense. 


40 


THE    ORIGINS 


This  idea  led  Jordamis  to  a  proof  of  the  rule  of  the  equilibrium  of  the 
straight  lever  whose  originality  cannot  be  contested. 


h    b 


Fig,  10 


46  Let  acb  be  the  beam,  a  and  b  the  weights  that  it  carries,  and  suppose 
that  the  relation  of  b  to  a  is  the  same  as  that  of  ca  to  c6.  I  maintain 
that  this  rule  will  not  change  its  place.  Indeed,  if  the  arm  supporting  b 
falls  and  the  beam  takes  up  the  position  c?ce,  the  weight  6  will  descend 
by  he  and  a  will  rise  by  fd.  If  a  weight  equal  to  the  weight  b  is  placed 
at  I,  at  a  distance  such  that  d  =  c6,  this  will  rise  in  the  motion  by 
gm  =  he.  But  it  is  clear  that  dfis  to  mg  as  the  weight  Z  is  to  the  weight  a. 
Consequently,  what  is  sufficient  to  bring  a  to  A  will  be  sufficient  to  bring 
1  to  m.  But  we  have  shown  that  b  and  I  counterbalance  each  other  exact 
ly,  so  that  the  supposed  motion  is  impossible.  This  will  also  be  true 
of  the  inverse  motion.  " 

Duhem  writes  in  this  connection  I — 

"  Underlying  this  demonstration  of  Jordanus  the  following  principle 
is  clearly  evident— that  which  can  lift  a  weight  to  a  certain  height  can 

also  lift  a  weight  which  is  k  times  as 
great  to  a  height  which  is  k  times  less. 
This  principle  is  then  the  same  as  that 
which  Descartes  took  as  a  basis  for  his 
complete  theory  of  statics  and  which, 
thanks  to  John  Bernoulli,  became  the 
principle  of  virtual  work.  " 

Jordanus  was  less  fortunate  when  he 
turned  his  attention  to  the  angular  lever. 
He  considered  a  lever  acf  carrying  equal 
weights  at  a  and /which  were  placed  in 
such  positions  that  ac  =  ef.  JordanuH 
was  of  the  opinion  that,  under  these 


Fig.  11 


1  o.  s.,  Vol.  r,  P.  121. 


THE    Xlllth   CENTURY  41 

conditions,  a  would  dominate  /.  He  arrived  at  this  conclusion  by£con- 
sidering  two  equal  arcs  al  and  mf.  It  is  clear  that  the  "  direct  "  taken 
by  the  weight  a  is  greater  than  the  "  direct  "  taken  by  the  weight  /. 
This  incorrect  conclusion  is  obtained  because,  since  the  linkages  are 
rigid,  the  two  displacements  al  andjfm  are  not  simultaneously  possible. 
Jordanus  thus  misunderstood  the  idea  of  moment. 

As  early  as  the  Xlllth  Century  the  Elementa  Jordani  were  generally 
united  by  the  copyists  with  De  Canonio  to  form  the  Liber  Euclidis  de 
ponderibus.*  This  artificial  associations  and  this  imaginative  titles 
are  the  despair  of  the  scholars  and  it  has  needed  all  the  learning  of 
Duhem  to  elucidate  them. 

Every  truly  novel  idea  evokes  a  reaction.  The  Elementa  Jordani 
did  not  provide  an  exception  to  this  rule.  In  the  Xlllth  Century  a 
critic  wrote  a  commentary  of  Jordanus  which  Duhem  calls  the  Peripa 
tetic  Commentary.*  This  author  at  every  turn  invokes  the  authority  of 
Aristotle  and  has  scruples  about  applying  the  gravitas  secundum  situm 
to  a  motionless  point  —  in  modern  language,  about  making  appeal  to 
a  virtual  displacement.  It  does  not  appease  his  conscience  to  consider 
that  rest  is  the  end  of  motion.  "  The  scientific  value  of  the  Commen 
tary  is  nothing,  "  declares  Duhem.3  "  But  its  influence  did  not 
disappear  for  a  very  long  time,  and  even  the  great  geometers  Tarta- 
glia,  Guido  Ubaldo  and  Mersenne  had  not  entirely  freed  themselves 
from  it.  " 

2.    TlIE    ANONYMOUS    AUTHOR    OF    "  LlBER    JORDANI    DK    RATIONE    PON- 
DERIS.  "      TlIE  ANGULAR  LEVKR.      TlIE  INCLINED  PLANE. 

We  now  come  to  an  especially  noteworthy  work  which  figures  in  the 
same  manuscript  as  the  Peripatetic  Commentary  under  the  title  Liber 
Jordani  de  ratione  pondrris,  and  which  did  not  remain  unknown  in  the 
Renaissance,  Tartaglia  sent  it  to  CurtiuH  Trojanus  who  published  it 
in  1565.  This  work  supereedes  and  rectifies  the  Klementa  Jordani  on 
many  important  points.  All  the  same,  it  is  based  on  the  same  principle 
of  gnivitas  secundum  situm. 

Duhem,  who  brought  this  manuscript  to  light,  terms  the  anonymous 
author  a  u  Precursor  of  Leonardo  da  Vinci.  "  Indeed,  in  many  respects 
this  precursor  surpassed  Leonardo,  who,  for  example,  spent  himself 
in  fruitless  efforts  to  evaluate  the  apparent  weight  of  a  body  on  an 
inclined  plane.  It  seems  more  natural  to  simply  speak  of  an  anonymous 


Nationals,  Paris,  latin  collection,  MHH.  7310  and  10,260, 

2  Ibid.,  Ms.  7378  A. 

3  0.  S.,  Vol.  I,  p.  13k 


42 


THE    ORIGINS 


author  of  the  Xlllth  Century,  a  disciple  of  Jordanus  who  had  out 
stripped  his  master. 

In  connection  with  the  bent  lever  this  author  corrected  Jordanus's 
error.  As  before,  let  a  lever  acf  carry  equal  weights  at  a  and  f  and 
be  placed  in  such  a  position  that  aaf  —  ff'. 


Fig.  12 


It  is  impossible  that  the  weight  a  should  dominate  the  weight  /. 
For  if  two  arcs  aft,  j#,  are  considered  on  the  two  circles  drawn  through 
a  and  /  and  corresponding  to  equal  angles  ach  and  jfr£  the  descent 
of  a  along  rh  necessitates  that  the  equal  weight  at /should  rise  through 
a  distance  In  which  is  greater  than  rh.  This  is  impossible. 

In  the  same  way  it  can  be  seen  that  /  will  not  dominate  a.     For 

if  the  arcs/E  and  am  correspond  to  equal 
angles  fcx  and  acm,  the  descent  of /along 
tx  makes  it  necessary  that  the  equal 
weight  placed  at  a  should  rise  by  pm, 
which  is  greater  than  tx.  This  is  impos 
sible.  Therefore  there  is  equilibrium  in 
the  position  considered,  in  which  aa'^ff'. 
The  anonymous  author  generalised 
this  result  to  an  angular  balance 
carrying  unequal  weights  at  a  and  6, 


Fig.  13 


THE    XHIth   CENTURY  43 

and  obtained  the  result  that  in  equilibrium  it  is  necessary  that  the 
distances  aaf  and  66'  from  a  and  6  to  the  vertical  drawn  through  the 
point  of  support,  c,  are  in  inverse  ratio  to  the  weights  a  and  6. 

We  see  that  this  author  knew  and  used  the  notion  of  moment. 
Elsewhere  he  wrote  on  this  subject,  "  If  a  load  is  lifted  and  the 
length  of  its  support  is  known,  it  can  be  determined  how  much 
this  load  weighs  in  all  positions.  The  weight  of  the  load  carried  at 
e  by  the  support  be  will  be  to  the  weight  carried  at  /  by  fb  as  el  is 
to  fr  or  as  pb  is  to  xb.  A  weight  placed  at  e,  at  the  end  of  the 
lever  6e,  will  weigh  as  if  it  were  at 
u  on  the  lever  6/.  " 

Thus  the  idea  of  gravitas  secundum 
situm,  which  Jordanus  had  used  qua 
litatively,  became  precise. 

Our  anonymous  author  also  con 
cerned  himself  with  the  stability  of 
the  balance,  and  rectified  certain 
errors  which  were  contained  in  the 
relevant  parts  of  Problems  of  Mecha 
nics. 


x  p  o 

More    than    this,    he    resolved    the 

problem  of  the  equilibrium  of  a  heavy  l^* 

body  on  an  inclined  plane,  a   problem 

which  had  eluded  the  wisdom  of  the  greek  and  alexandrian  geometers. 

In  order  that  this  may  be  done,  it  is  first  observed  that  the  gravitas 
secundum  situm  of  a  weight  on  an  inclined  plane  is  independent  of 
its  position  on  the  plane.  The  author  then  attempts  a  comparison  of 
the  value  that  that  gravity  takes  on  differently  inclined  planes.  We 
shall  quote  from  Duhetn's  translation  of  this  same  Xlllth  Century 
manuscript. 

u  If  two  weights  descend  by  differently  inclined  paths,  and  if  they 
are  directly  proportional  to  the  declinations,  they  will  be  of  the  same 
strength  in  their  descent. 

"  Let  ab  be  a  horizontal  and  W,  a  vertical.  Suppose  that  two 
oblique  lines  da  and  dc  fall  on  one  side  and  on  the  other  of  6rf,  and 
that  dc  IISLS  the  greater  relative  obliquity.  By  the  relation  of  the  obli 
quities  I  understand  the  relation  of  the  declinations  and  not  the  relation 
of  the  angles  ;  this  means  the  relation  of  the  lengths  of  the  named 
lines  counted  as  far  as  their  intersection  with  the  horizontal,  in  such 
a  way  that  they  take  simUarily  of  the  direct. 

**  In  the  second  place,  let  e  and  h  be  the  weights  placed  on  dc  and 
da  respectively,  and  suppose  that  the  weight  e  is  to  the  weight  h  as 


44  THE    ORIGINS 


dc  is  to  da.  I  maintain  that  in  such  a  situation  the  two  weights  will 
have  the  same  strength. 

"  Indeed,  let  dk  be  a  line  having  the  same  obliquity  as  dc  and, 
on  that  line,  let  there  be  a  weight  g  which  is  equal  to  e. 

"  Suppose  that  the  weight  e  should  descend  to  Z,  if  that  is  possible, 
and  that  it  should  draw  the  weight  h  to  m.  (It  is  clear  that  the  author 
imagines  the  two  weights  to  be  connected  by  a  thread  which  passes 
over  a  pulley  at  d.)  Take  gn  equal  to  Am,  and  consequently  equal 
to  el.  Draw  a  perpendicular  to  db  which  passes  through  g  and  A, 
say  ghy.  Drop  a  perpendicular  It  from  the  point  I  onto  db.  Then 
drop  [the  perpendiculars]  nr,  mx,  and  ez.  The  relation  of  nr  to  ng  is 
that  of  dy  to  dg  and  also  that  of  db  to  dk.  Therefore  mx  is  to  nr  as 
dk  is  to  da  ;  that  is  to  say,  as  the  weight  g  is  to  the  weight  h.  But  as  c 
is  not  able  to  pull  g  up  to  n  (nr  =  ez),  it  is  no  better  able  to  pull  h  up 
to  m.  The  weights  therefore  remain  in  equilibrium.  " 

This  demonstration,  which  leads  to  the  correct  law  of  the  apparent 
heaviness  of  a  body  on  an  inclined  plane,  was  directly  inspired  by  that 
of  Jordanus  concerning  the  equilibrium  of  a  straight  lever*  Like 
that,  it  implicitly  proceeds  according  to  the  principle  of  virtual  work. 

We  shall  now  give  some  indication  of  the  ideas  on  dynamics  which 
were  used  by  the  author  of  Liber  Jordani  de  ratione  pondcris. 

The  environment's  resistance  to  the  motion  of  a  body  depends 
on  the  shape  of  the  body,  which  penetrates  the  environment  the  better 
as  its  shape  is  the  more  acute  and  its  figure  the  more  smooth.  It 
depends,  in  the  second  place,  on  the  density  of  the  fluid  traversed.  AH 
media  are  compressible ;  the  lower  strata,  compressed  by  the  upper 
ones,  are  the  denser  and  those  which  hinder  motions  more.  At  the 
front  of  the  moving  body  will  be  a  part  of  the  medium  compressed 
on,  and  sticking  to  it.  But  the  other  parts  of  the  medium,  which 
are  displaced  by  the  moving  body,  curl  round  behind  to  occupy  the 


THE    XHIth   CENTURY  45 

space  which  the  body  has  left  empty.  This  motion  of  lateral  parts 
of  the  medium  may  be  compared  to  the  bending  of  an  arc. 

The  more  heavy  the  medium  is  at  traversal,  the  slower  is  the  des 
cent  of  a  heavy  body. 

The  descent  is  slower  in  a  fluid  which  is  more  dense. 

Greater  width  diminishes   the   gravity. 

A  heavy  thing  will  move  more  freely  as  the  duration  of  its  fall  in 
creases.  "  This  is  more  true  in  air  than  in  water,  because  air  is  suited 
to  all  kinds  of  motion.  Thus  a  falling  body  drags  with  it,  from  the  outset 
of  its  motion,  the  fluid  that  lies  behind  it  and  sets  in  motion  the  fluid 
in  its  immediate  contact.  The  parts  of  the  medium  set  in  motion 
in  this  way,  in  their  turn  move  those  that  adjoin  them,  in  such  a  way 
that  the  latter,  which  are  already  in  motion,  present  a  lesser  obstacle 
to  the  falling  body.  For  this  reason  the  body  becomes  heavier  and 
imparts  a  greater  impulsion  to  the  parts  of  the  medium  which  it  dis 
places  until  these  are  no  longer  simply  pushed  by  the  body,  but  drag  it 
along  with  them.  Thus  it  happens  that  the  gravity  of  a  moving 
body  is  increased  by  their  traction  and  that,  reciprocally,  their  motion 
is  multiplied  by  this  gravity  so  that  it  continually  increases  the  velocity 
of  the  body. " 

The  shape  of  a  heavy  body  affects  the  strength  of  its  weight. 

The  strength  of  a  motive  agency  seems  to  be  equally  baulked  by 
a  body's  very  large  or  very  small  weight. 

Rotation  of  a  propellant  increases  its  strength,  and  does  so  more 
effectively  as  the  radius  is  greater. 

A  body  whose  parts  are  coherent  is  thrown  directly  backwards  if 
it  is  stopped  by  a  collision  during  its  motion.  "  The  parts  of  a  moving 
body  A  that  He  in  front  are  the  first  to  meet  the  obstacle  C.  They 
are  therefore  compressed  by  the  mass  and  the  impetuosity  of  the  parts 
which  lie  behind  them,  and  are  forced  to  condense.  The  impetuosity 
of  the  parts  behind  is  annulled  in  this  way.  The  parts  in  front  now 
assume  their  original  volume  again  and  recoil  backwards,  thus  com 
municating  an  impulsion  to  the  others.  If  the  parts  which  are  com 
pressed  in  this  way  are  able  to  detach  themselves  from  each  other 
they  will  be  thrown  off  in  one  direction  and  another.  ** 

If  the  heaviness  of  a  body  is  not  uniform,  the  denser  part  will 
place  itself  in  front,  whatever  the  part  to  which  the  impulsion  is  given.1 

These  ideas  on  dynamics  held  by  Jordanus's  School  are  much 
less  interesting  and  moreover,  less  original  than  its  statics.  We  have 
cited  them  here  as  curiosities. 

1  Cf.,  DUHEM,  £tude$  sur  Leonard  de  Vinci^  Series  I  (Hermann),  1906* 


46  THE    ORIGINS 

From  the  historical  point  of  view  it  must  be  remarked  that  Duhem, 
in  writing  his  first  studies  on  the  origin  of  statics,  first  believed  the 
work  of  this  unknown  disciple  of  Jordanus  to  be  entirely  original. 
But  the  later  discovery  of  a  XHIth  Century  manuscript1  led  him  to 
a  treatise  De  Ponderibus  which  was  more  complete  than  the  Liber 
Jordani  de  ratione  ponderis. 

Now  this  treatise,  divided  into  four  books,  seems  to  be  a  complex 
in  which  various  works  have  been  artificially  united.  There  is  first 
a  book,  of  indisputable  Medieval  origin,  that  repeats  the  demonstra 
tions  Jordanus  used  and  supplements  them  by  the  condition  for  the 
equilibrium  of  the  angular  lever  and  the  determination  of  the  apparent 
weight  of  a  body  on  an  inclined  plane.  The  second  book  appears 
to  have  been  inspired  by  De  Canonio  while  the  third  treats  the  concept 
of  moment  and  the  conditions  for  the  stability  of  the  balance.  Finally 
there  is  a  fourth  book  devoted  to  dynamics. 

The  last  two  books  are  closely  related  to  Problems  of  Mechanics 
although  they  alter,  correct  and  complete  this  work  in  many  places. 
Certain  indications  led  Duhem  to  surmise  that  the  two  books  are  a 
relic  of  greek  science  and  were  probably  handed  on  by  the  Arabs — 
this  because  no  latinised  greek  terms  are  found  in  them.2  Accord 
ingly  it  is  possible  that  our  unknown  author  did  not  discover  the 
idea  of  moment  himself.  This  limits  the  originality  of  his  work,  but 
it  remains  that  gravitas  secundum  situm  properly  belongs  to  the  Xlllth 
Century  School,  and  that  it  was  used  by  this  School  to  obtain  a  correct 
solution  of  the  problem  of  the  inclined  plane  long  before  Stevin  and 
Galileo  did  so, 

1  Biblioth&que  Nationale,  latin  collection,  Ms.  8680  A. 

2  C/.  DUHEM,  0.  S.,  Vol.  II,  note  F,  p.  318.    LEONARDO  DA  VINCI  himself  seema  to 
have  been  unaware  of  the  three  last  books  of  De  Poncferi&us— another  argument  for  not 
regarding  the  unknown  author  as  his  precursor. 


CHAPTER   FOUR 

THE  XlVth  CENTURY 

THE  SCHOOLS  OF  BURIDAN  AND  ALBERT  OF  SAXONY 
NICOLE  ORESME  AND  THE  OXFORD  SCHOOL 


1.   THE  DOCTRINE   OF  "  IMPETUS  "    (JOHN 

The  idea  of  attributing  a  certain  energy  to  a  moving  body  solely 
on  account  of  its  motion  is  entirely  foreign  to  aristotelian  dynamics. 

In  Antiquity  John  of  Alexandria — surnamed  Philopon — who  lived 
in  the  Vth  Century  A.  C.,  was  alone  in  disputing  Aristotle's  belief  in 
this  matter.  Thus  he  held  that  the  air  which  was  set  in  motion  could 
not  become  the  motive  agency  of  a  projectile,  whose  motion  was, 
on  the  contrary,  easier  in  a  vacuum  than  in  air.  "  Whoever  throws  a 
projectile  embodies  in  it  a  certain  action,  a  certain  power  of  self-move 
ment  which  is  incorporated.  .  .  .  Nothing  prevents  a  man  from  throwing 
a  stone  or  an  arrow  even  when  there  is  no  other  medium  than  the  vacuum. 
A  medium  hinders  the  motion  of  projectiles,  which  cannot  advance 
without  dividing  it — nevertheless  they  can  move  through  these  media* 
Nothing  therefore  prevents  an  arrow,  a  stone  or  any  other  body  from 
being  thrown  in  the  vacuum.  Indeed,  the  motive  agency,  the  moving 
body  and  the  space  that  will  receive  the  projectile  are  all  present."1 

Philopon's  thesis  was  handed  on  to  the  Middle  Ages  by  the  Arabs 
— in  particular,  by  the  astronomer  Al  Bitrogi,  But  while  assuming  the 
existence  of  a  "  property  which  remains  attached  to  a  stone  or  an  arrow 
after  the  projectile  has  been  thrown,  "  he  held  that  this  property 
decreased  at  such  a  rate  and  to  such  an  extent  as  the  projectile  was 
separated  from  its  motive  agency. 

Albertus  Magnus  and  Saint  Thomas  Aquinas  knew  of  this  tradition 
but  did  not  give  the  least  credit  to  John  Philopon's  argument.  For 
example,  Saint  Thomas  Aquinas  believed  that  if  the  existence  of  an 

1  Erudissima  commentaria  in  prim&s  quatuor  Aristotelis  de  naturali  auscultations 
libros*  Venice  (1532),  Trans.  DUHEM, 


48  THE    ORIGINS 

apparent  property  impressed  on  a  moving  body  were  assumed,  "  violent 
motion  would  arise  from  an  intrinsic  property  of  a  moving  body,  which 
is  contrary  to  the  very  notion  of  violent  motion.  Moreover,  it  would 
follow  from  this  that  a  stone  would  be  altered  in  its  substantial  form 
by  the  very  fact  that  it  moved  from  place  to  place,  which  is  contrary 
to  common  sense.  "  I 

Roger  Bacon,  Walter  Burley  and  John  of  Jandun  all  adopted 
Aristotle's  doctrine  on  this  matter.  The  first  Schoolman  to  oppose 
this  opinion  was  William  of  Ockham  (1300-1350).  He  asked  himself 
where  the  motive  agency  might  be.  It  cannot  reside  in  the  apparatus 
or  organism  that  has  thrown  the  projectile,  for  this  apparatus  can  be 
destroyed  immediately  after  the  launching  without  interupting  the  pro 
gress  of  the  projectile. 

Nor  can  the  motive  agency  be  the  air  which  is  set  in  motion.  For 
the  arrows  of  two  archers  which  are  shot  towards  each  other  can  be 
arranged  to  collide  with  each  other,  which  requires  that  the  same  air 
produces  two  different  motions  at  the  same  time. 

There  cannot  be  distinguished  elsewhere  a  cause  that  could  provide 
the  motive  power.  Such  a  cause  cannot  reside  in  the  launching  appa 
ratus  nor  in  the  motion  of  the  projectile  itself.  If  something  which  is 
its  own  motive  agency  is  thrown,  that  which  moves  the  body  cannot  be 
distinguished  from  the  moving  body  itself.  Moreover,  motion  from 
place  to  place  is  not  something  which  is  renewed  at  each  instant,  requir 
ing  the  constant  presence  of  a  motive  cause.  It  is  true  that  the  pro 
jectile  passes  through  a  different  region  at  each  instant,  but  this  does  not 
in  itself  constitute  anything  novel.  It  is  only  novel  with  respect  to  the 
moving  body.2 

Thus  William  of  Ockham  decided  to  reject  Aristotle's  axiom  which 
requires  the  continuous  existence  of  a  motive  agency  in  contact  with, 
yet  not  part  of,  the  projectile.  He  did  not,  however,  replace  it  by  any 
new  principle. 

We  now  arrive  at  the  doctrine  of  impetus  that  was  conceived  by 
Buridan.  John  I.  Buridan,  of  Bethune,  was  rector  of  the  University  of 
Paris  in  1327  and  canon  of  Arras  in  1342.  He  died  in  Paris  after  1358.:i 

In  a  memoir  called  Quaestiones  octavi  libri  physicorum^   Buridan 

1  Opera  omnia,  Vol.  Ill — Commentaria  in  libros  Aristotclis  de  Caclo  ft  Mundo* 
Book  III,  lect.  VII. 

2  Cf.  DUBDEM,  Etudes  sur  Leonard  de  Vinci,  Series  II,  p.  192. 

3  DUHEM,  who  has  studied  BURIDAN'S  works  in  detail,  including  those  concerning 
free  will,  says  that  he  has  found  no  trace  of  the  parable  of  the  ass,  which  apart  from  IUH 
status  in  the  history  of  mechanics,  has  made  Buridan's  name  classical. 

4  BiUiotheque  Nationale,  Paris,  latin  collection,  Ms.  14,723,  fol  106-107.     In  the 
text  we  are  following  DUHEM'S  translation. 


THE    XlVth    CENTURY  49 

discussed  the  scholastic  thesis  of  the  motion  of  projectiles.  Aristotle, 
he  says,  mentions  two  opinions  on  this  matter. 

The  first  invokes  &vTineQiaTcx.ai,<;.  As  a  projectile  moves  rapidly 
away  from  its  position,  Nature,  who  does  not  allow  the  existence  of  a 
vacuum,  makes  the  air  behind  the  projectile  rush  in  towards  this  position 
with  the  same  velocity.  This  air  pushes  the  projectile  and  the  same 
effect  is  reproduced,  at  least  for  a  certain  distance.  This  opinion  is 
rejected  by  Aristotle — if  no  other  principle  than  &.VT metier &oi<;  is 
invoked,  it  is  necessary  that  all  bodies  which  are  behind  the  particle, 
including  the  sky  itself,  participate  in  the  projectile's  motion.  Indeed, 
the  air  will  also  leave  its  position.  It  is  then  necessary  that  another 
body  must  replace  it,  and  so  on  in  an  indefinite  sequence,  unless  it  is 
assumed  that  a  certain  rarefaction  of  bodies  behind  the  projectile  is 
produced. 

According  to  the  second  opinion,  which  Aristotle  seems  to  have 
supported,  the  launching  of  a  projectile  disturbs  the  ambient  air  at  the 
same  time.  This  air,  violently  set  in  motion,  has,  in  its  turn,  the  power 
to  move  the  projectile.  The  first  mass  of  air  moves  the  projectile  until 
this  comes  to  a  second  mass  of  air.  This  second  one  moves  it  until  a 
third  is  reached,  and  so  on.  Further,  Aristotle  is  heard  to  say,  there 
is  not  merely  a  single  moving  body,  but  sxiccessive  moving  bodies,  a 
series  of  consecutive  or  contiguous  motions. 

Buridan  set  the  following  observations  against  these  theories.  A  top 
or  a  grindstone  will  turn  for  a  very  long  time  without  leaving  its  position, 
in  such  a  way  that  the  air  does  not  have  to  follow  it  to  fill  an  abandoned 
place.  Further,  the  wheel  will  continue  to  txirn  if  it  is  covered  and  thus 
separated  from  the  surrounding  air.  A  javelin  whose  following  end  is 
armed  with  a  point  as  sharp  as  its  tip  will  move  as  rapidly  as  if  this  were 
not  tapered  at  the  back.  Now  since  the  air  is  easily  divided  by  the 
javelin's  sharpness,  it  cannot  push  strongly  on  this  backward  pointed 
part.  A  ship  will  continue  to  move  for  a  long  time  after  towing  has 
been  stopped,  and  a  boatman  will  not  feel  the  air  pushing  it — on  the 
contrary,  he  feels  the  air  slowing  down  the  ship's  motion. 

The  air  set  in  motion  should  be  able  to  move  a  feather  more  easily 
than  a  stone.  Now  we  are  not  able  to  throw  a  feather  as  far  as  a  stone. 

Buridan  himself  put  forward  the  following  thesis. 

"  Whenever  some  agency  sets  a  body  in  motion,  it  imparts  to  it  a 
certain  impetus,  a  certain  power  which  is  able  to  move  the  body  along 
in  the  direction  imposed  upon  it  at  the  outset,  whether  thia  be  upwards, 
downwards,  to  the  side  or  in  a  circle.  The  greater  the  velocity  that  the 
body  is  given  by  the  motive  agency,  the  more  powerful  will  be  the 
impetus  which  is  given  to  it.  It  is  this  impetus  which  moves  a  stone 


50  THE    ORIGINS 

after  it  lias  been  thrown  until  the  motion  is  at  an  end.  But  because 
of  the  resistance  of  the  air  and  also  because  of  the  heaviness,  which 
inclines  the  motion  of  the  stone  in  a  direction  different  from  that  in 
which  the  impetus  is  effective,  this  impetus  continually  decreases.  Con 
sequently  the  motion  of  the  stone  slows  down  without  interruption. 
Finally  the  impetus  is  overcome  and  destroyed  at  the  point  where 
gravity  dominates  it,  and  henceforth  the  latter  moves  the  stone  towards 
its  natural  place.  .  .  . 

"  All  natural  forms  and  dispositions  are  received  by  matter  in  pro 
portion  to  itself.  Consequently  the  more  matter  a  body  contains,  the 
more  impetus  can  be  imparted  to  it,  and  the  greater  is  the  intensity  with 
which  it  can  receive  the  impetus.  ...  A  feather  receives  such  a  weak 
impetus  that  this  is  immediately  destroyed  by  the  resistance  of  the  air. 
In  the  same  way,  if  someone  throws  projectiles  and  sets  in  motion  with 
equal  velocities  a  piece  of  wood  and  a  piece  of  iron,  which  have  the  same 
volume  and  the  same  shape,  the  piece  of  iron  will  travel  further  because 
the  impetus  which  is  imparted  to  it  is  stronger.  It  is  for  the  same  reason 
that  it  is  more  difficult  to  stop  a  large  blacksmith's  wheel,  moving 
rapidly,  than  a  smaller  one.  ..." 

In  short  the  impetus,  in  Buridan's  sense,  increases  with  the  velocity. 
In  addition,  it  is  proportional  to  the  density  and  to  the  volume  of  the 
body  concerned.  Further,  in  Buraidan's  view,  the  existence  of  impetus 
explained  the  acceleration  of  falling  bodies. 

"  The  existence  of  impetus  seems  to  be  the  cause  by  which  the  natural 
fall  of  bodies  accelerates  indefinitely.  At  the  beginning  of  the  fall, 
indeed,  the  body  is  moved  by  gravity  alone.  Therefore  it  falls  more 
slowly.  But  before  long  this  gravity  imparts  a  certain  impetus  to  the 
heavy  body — an  impetus  which  is  effective  in  moving  the  body  at  the 
same  time  as  gravity  does.  Therefore  the  motion  becomes  more  rapid. 
But  the  more  rapid  it  becomes,  the  more  intense  the  impetus  becomes. 
Therefore  it  can  be  seen  that  the  motion  will  be  accelerated  con 
tinuously.  " 

Further,  Buridan  applied  the  notion  of  impetus  to  stars  as  well  as 
to  terrestrial  bodies. 

46  In  the  Bible  there  is  no  evidence  of  the  existence  of  intelligences 
charged  with  communicating  their  appropriate  motion  to  the  heavenly 
bodies.  It  is  therefore  permissible  to  show  that  there  is  no  necessity  to 
suppose  the  existence  of  such  intelligences.  Indeed  it  can  be  said  that 
when  He  created  the  World,  God  set  each  of  the  heavenly  bodies  in 
motion  in  the  way  that  he  had  chosen — that  He  imparted  to  each  of 
them  an  impetus  which  has  kept  it  moving  ever  since.  Thus  God  no 
longer  has  to  move  these  bodies,  except  for  a  general  influence  similar 


THE    XlVth    CENTURY  51 

to  that  by  "which  He  gives  his  consent  to  all  things  that  occur.  It  is  for 
this  reason  that,  on  the  seventh  day,  He  was  able  to  rest  from  the  tasks 
which  He  had  accomplished  and  to  confine  himself  to  the  creation  of 
things  concerning  mutual  actions  and  feelings.  The  impetus  that  God 
imparted  to  the  heavenly  bodies  is  neither  weakened  nor  destroyed  by 
the  passage  of  time.  For  in  these  heavenly  bodies  there  are  no  ten 
dencies  towards  other  motions  and  because,  moreover,  there  is  no  longer 
any  resistance  which  could  corrupt  and  repress  this  impetus.  I  do  not 
say  all  this  with  complete  assurance.  I  would  only  ask  the  theologians 
to  show  me  how  all  these  things  happen.  " 

As  a  true  Scholastic  Buridan  believed  himself  obliged  to  defend  the 
doctrine  of  impetus  from  the  metaphysical  objections  that  could  be 
advanced  against  it.  The  motion  of  a  projectile  is  a  violent  one  in 
Aristotle's  sense.  Now,  according  to  the  Ethics  (Book  III),  violent 
phenomena  stem  from  an  extrinsic,  not  an  intrinsic,  cause.  To  this 
Buridan  replied  that  the  impetus  of  a  moving  body  is  effectively  violent, 
not  natural.  The  nature  of  heavy  things  favours  a  different  motion 
and  the  destruction  of  the  impetus.  On  the  question  of  whether  the 
impetus  is  distinct  from  the  motion  and  whether  it  is  of  a  permanent 
kind,  Buridan  replied  that  impetus  could  not  itself  be  motion  because 
all  motion  requires  a  motive  agency  ;  that  impetus  was  a  permanent 
reality,  distinct  from  the  local  motion  of  the  projectile  ;  and  that  it  was 
probable  that  the  impetus  was  a  quality  whose  nature  was  to  actuate  the 
body  to  which  it  was  imparted. 

These  subtleties  add  nothing  to  Buridan's  positive  doctrine.  It  is 
more  important  to  remark  that  Buridan  maintained  that  the  impetus 
lasted  indefinitely  if  it  was  not  diminished  by  a  resistance  of  the  medium 
or  modified  by  some  agency  affecting  the  moving  body.  This  is  the 
germ  of  the  modern  principle  of  inertia. 

2.   TlIE    SPHERICITY    OF    THE    EARTH    AND    THE     OCEANS— ALBERT     OF 

SAXONY  AND  THE  ARLSTOTLETIAN  TRADITION.  H 

In  the  first  chapter  of  this  book  we  referred  to  the  a  priori,  or  physical 
proofs,  and  the  a  posteriori  proofs  which  Aristotle  gave  of  the  sphericity 
of  the  Earth  and  the  oceans.  For  better  or  worse,  tradition  preserved 
and  enriched  these  proofs. 

Pliny  the  Elder,  in  his  Natural  History,  supplemented  Aristotle's 
evidence  with  facts  that  strictly  derive  from  capillarity — the  sphericity 
of  drops  of  water,  the  meniscuses  of  liquids,  etc.  .  * .  Ptolemy  only 
retained  the  a  posteriori  proofs  which  Aristotle  had  given.  Simplicius, 
in  his  commentary  on  De  CaeJo,  corrected  the  dimensions  attributed  to 


52  THE   ORIGINS 

the  Earth  after  Erastosthenes'  evaluation.     Averroes  confined  himself 
to  an  elaboration  of  Aristotle's  evidence. 

John  Sacro  Bosco — the  author  of  a  treatise  called  De  Sphaera  which 
became  the  most  widely  known  cosmography  in  the  Xlllth  Century — 
reproduced  Ptolemy's  account.  Albertus  Magnus  firmly  excluded  the 
evidence  depending  on  the  sphericity  of  water  drops.  Saint  Thomas 
Aquinas  limited  himself  to  Aristotle's  proofs  alone,  while  Roger  Bacon 
supplemented  them  with  the  following  corollary  which  was  acclaimed 
by  the  Schoolmen — any  given  vessel  will  contain  a  smaller  quantity 
of  liquid  as  it  is  taken  further  from  the  centre  of  the  Earth. 

We  now  come  to  Albert  of  Rickmersdorf,  called  Albert  of  Saxony. 
Though  his  biography  is  somewhat  mysterious,  it  is  certain  that  he  was 
enlisted  at  the  Sorbonne  from  1350  to  1361  and  that  he  was  rector  of 
the  University  of  Paris  from  1353.  His  Acutissimae  Quaestiones  on 
Aristotle's  Physics  had  considerable  repercussions,  and  his  influence  was 
felt  by  most  students  of  mechanics,  including  Galileo  himself. 

Albert  of  Saxony  suggested  going  back  to  the  measurement  of  a 
degree  of  meridian  at  different  latitudes  in  order  to  determine  the  true 
figure  of  the  Earth.  (This  idea  was  applied  by  John  Femel  at  the 
beginning  of  the  XVIth  Century  and,  of  necessity,  repeated  in  the 
XVIIth  Century.)  "  If  these  two  paths  are  found  to  be  equal  this  is 
a  certain  indication  that  the  Earth  is  circular  from  north  to  south.  If 
on  the  contrary,  it  were  found  that  they  lacked  equality  this  would  be 
an  indication  that  the  Earth  was  not  round  from  north  to  south.  " 

Like  Albertus  Magnus,  Albert  of  Saxony  excluded  the  evidence  pro 
vided  by  small  drops,  which  is  common  to  all  liquids,  like  mercury,  and 
is  especially  noticeable  in  small  quantities. 

Albert  of  Saxony  stated  the  following  corollaries,  which  were  to 
become  popular  among  the  Schoolmen. 

"  1.  From  the  fact  that  the  Earth  is  round  it  follows  that  lines  normal 
to  the  surface  of  the  Earth  will  approach  each  other  continuously,  and 
meet  at  the  centre. 

"2.  It  follows  that  if  two  vertical  towers  are  built,  the  higher  they 
become,  the  further  away  from  each  other  they  will  be  ;  and  that  the 
deeper  they  are,  the  nearer  together  they  will  be. 

"  3.  If  a  well  is  dug  with  a  plumb-line,  it  will  be  larger  near  the 
opening  than  at  the  bottom. 

"  4.  Any  line  such  that  all  its  points  are  at  an  equal  distance  front 
the  centre  is  a  curved  line.  If  a  straight  line  touches  the  Earth's  surface 
at  its  middle  point,  this  point  will  be  nearer  to  the  centre  than  the  ends 
of  the  line.  It  follows  that  if  a  man  goes  along  this  straight  line,  he 


THE    XFVth   CENTURY  53 

descend  for  a  time  and  then  will  rise.  He  will  descend,  indeed, 
until  lie  has  come  to  the  point  which  is  nearest  to  the  centre  of  the 
Earth  and  will  rise  from  the  moment  that  he  leaves  that  point  behind 
him. 

"  From  this  it  can  be  concluded  that  a  body  which  describes  a  tra 
jectory  between  two  fixed  ends,  a  trajectory  which  either  rises  or  falls 
without  interruption,  must  necessarily  travel  a  shorter  distance  than  if 
the  path  went  from  one  point  to  the  other  without  rising  or  falling.  This 
is  seen  clearly  if  it  is  supposed  that  the  first  trajectory  is  a  diameter  of 
the  Earth  and  the  second  is  a  half- circumference  having  this  diameter 
as  chord. 

"  5.  When  a  man  walks  on  the  surface  of  the  Earth  his  head  moves 
more  quickly  than  his  feet.  .  .  .  One  can  conceive  of  a  man  so  tall  that 
his  head  moves  in  the  air  twice  as  quickly  as  his  feet  move  over  the 
ground.  " 

These  paradoxes  are  typical  of  the  scholastic  attitude  of  mind  and  it 
is  for  this  reason  that  we  have  quoted  them.  Undoubtedly  they  were 
intended  to  stimulate  the  minds  of  students,  and  perhaps,  too,  to  confuse 
those  who  were  not  scholars.  The  dialectic  of  the  Schoolmen  was  not 
in  the  least  concerned  with  orders  of  magnitude.  It  was  amusing  to 
proliferate  the  consequences  of  the  convergence  of  verticals  and  their 
practical  parallelism  was  of  no  concern — that  was  a  notion  suitable  for 
craftsmen.  And  these,  in  their  turn,  were  not  much  worried  by  the 
comments  of  the  Schoolmen  when  they  were  building  their  towers  and 
digging  their  wells  according  to  the  simple  rules  of  their  practice. 

3,  ALBERT  OF  SAXONY'S  THEORY  OF  CENTRE  OF  GRAVITY. 

When  commenting  on  that  thesis  of  Aristotle  according  to  which 
there  exists,  in  each  heavy  body,  a  centre  of  gravity  (to  IJL&GQV)  which 
tends  to  be  carried  towards  the  centre  of  the  Universe,  Albert  of  Saxony 
specified  that  "  each  of  the  parts  of  a  heavy  body  is  not  moved  in  such 
a  way  that  its  own  centre  would  come  to  the  centre  of  the  World,  for 
this  would  be  impossible.  Rather  it  is  the  whole  body  which  falls  in 
such  a  way  that  its  centre  would  become  the  centre  of  the  World.  It  is 
false,  and  contrary  to  observation,  to  say  that  a  large  body  falls  more 
slowly  than  a  lighter  body,  or  that  ten  stones  which  are  united  together 
hinder  each  other's  fall.  " 

The  Earth,  limited  partly  by  the  concave  surface  of  the  water  and 
partly  by  the  concave  surface  of  the  air,  is  in  ita  natural  position  when 
its  centre  of  gravity  is  at  the  centre  of  the  World.  If  this  is  not  so,  it 


54  THE    ORIGINS 

will  start  falling  and  will  move  until  the  centre  of  the  aggregate  which 
it  forms  with  all  the  other  heavy  bodies  becomes  the  centre  of  the 
World. 

It  should  be  remarked,  as  Jouguet  has  done  in  this  connection,1 
that  Albert  of  Saxony's  concept  of  centre  of  gravity,  the  point  of  a  body 
at  which  all  the  weight  appears  to  be  concentrated,  was  a  purely  experi 
mental  one,  to  Mm  and  his  School.  It  was  not  the  same  as  the  modern 
conception  of  centre  of  gravity,  which  depends  on  the  approximation 
that  verticals  are  parallel.  On  the  contrary,  it  was  developed  together 
with  a  systematic  consideration  of  the  convergence  of  verticals  which 
was  carried  to  the  point  of  paradox,  as  we  have  seen.  This  co-existence 
was  at  the  root  of  several  fallacies  which  were  to  perplex  people,  even 
such  eminent  ones  as  Fermat,  until  the  XVIIth  Century.2 

We  return  to  Albert  of  Saxony.  Suppose  that  the  Earth  is  displaced 
from  its  natural  place — for  example,  to  the  concavity  of  the  orbit  of  the 
Moon — and  held  there  by  force.  Suppose  that,  there,  a  heavy  body  is 
allowed  to  fall.  Then  this  body  will  be  attracted  towards  the  centre  of 
the  World,  not  towards  the  centre  of  the  Earth.  "  If  heavy  bodies 
move  towards  the  ground,  this  is  in  no  way  caused  by  the  Earth,  but 
happens  because  they  approach  the  centre  of  the  World  by  going  to 
wards  the  Earth.  " 

The  Earth  does  not  have  a  uniform  gravity — "  the  part  which  is  not 
covered  by  sea,  being  exposed  to  the  rays  of  the  Sun,  is  more  dilated 
than  the  part  the  waters  cover.  Besides,  if  its  geometrical  centre  were 
to  coincide  with  its  centre  of  gravity,  and  consequently  with  the  centre 
of  the  World,  it  would  be  entirely  covered  by  the  waters.  " 

Here,  in  Albert  of  Saxony's  writings,  is  the  trace  of  an  argument  that 
had  preoccupied  some  of  his  immediate  predecessors.  If  all  elements, 
declared  Walter  Burley  (1275-1357),  had  the  form  of  spheres  with  centres 
at  the  centre  of  the  Universe,  each  would  be  in  its  natural  place —  but 
then  the  Earth  would  be  completely  covered  with  water.  John  Duns 
Scot  (1275-1308)  resolved  this  difficulty,  in  his  Doctor  Subtilis,  with  a 
finalist  explanation — to  witt,  a  part  of  the  Earth  is  uncovered  with  a 
view  to  the  safety  of  living  beings. 

Albert  of  Saxony  believed  therefore  that  it  was  the  Earth's  centre, 
of  gravity,  not  its  geometrical  centre,  that  was  placed  at  the  centre  of 
the  World.  Furthermore,  the  Earth  was  not  fixed  in  position.  A 
host  of  reasons,  such  as  heating  by  rays  of  the  Sun,  could  produce  a 
variation  of  the  distribution  of  gravity  in  the  terrestrial  mass,  and  could 

1  JOUGUET,  L.  M.,  Vol.  I,  p.  60. 

2  This  question  was  at  the  root  of  the  controversy  on  Geostatics,  to  which  we 
shall  return. 


THE   XlVth   CENTURY  55 

displace  its  centre  of  gravity.  As  a  more  substantial  mechanism,  Albert 
of  Saxony  mentioned  erosion. 

The  question  arose  as  to  how  the  mass  of  the  waters  could  be  intro 
duced.  On  this  point  Albert  of  Saxony's  opinion  was  somewhat  variable. 

In  commenting  on  the  Physics  he  wrote,  "  What  I  have  written 
about  the  Earth  alone  may  be  understood  equally  for  the  whole  aggre 
gate  formed  by  the  Earth  and  the  waters.  These  two  elements  undoub 
tedly  form  a  total  and  unique  gravity  whose  centre  of  gravity  is  at  the 
centre  of  the  World.  "  At  this  same  centre  of  the  World  was  also  to 
be  found  the  centre  of  lightness  of  light  bodies. 

It  is  this  that  explains  the  following  picture  which  he  boldly  painted. 

"  Since  the  cold  is  especially  intense  at  the  poles,  the  layer  of  igneous 
element  there  must  be  thinner  than  at  the  equator  if  fire,  which  is  con 
tinuously  created  at  the  equator  is  not  to  run  towards  the  poles.  In 
the  same  way  that  water  constantly  runs  towards  lower  places  in  order 
that  the  centre  of  all  gravity  shall  be  at  the  centre  of  the  World,  so  we 
must  assume  that  fire  travels,  without  interruption,  from  the  equator 
towards  the  poles  in  order  that  its  centre  of  lightness  shall  be  at  the 
centre  of  the  World. 

"  It  should  be  imagined  that,  at  the  poles,  fire  is  constantly  being 
transformed  into  air,  and  at  the  equator  air  is  constantly  being  trans 
formed  into  fire  ;  and  that  fire  continually  runs  from  the  equator  to 
wards  the  poles  in  order  that  the  centre  of  all  lightness,  like  the  centre 
of  gravity,  shall  be  found  at  the  centre  of  the  World.  " 

In  short,  as  Duhem  has  observed,1  "the  common  centre  of  heavy 
bodies — both  the  closed  earth  and  the  water — and  the  common  centre 
of  light  bodies — both  air  and  fire — are  placed  at  the  centre  of  the 
World.  " 

However,  in  commenting  on  De  Caelo,  Albert  of  Saxony  took  a 
different  view. 

"  We  reply  by  denying  that  the  centre  of  the  World  coincides  with 
the  centre  of  the  aggregate  formed  by  the  earth  and  the  water.  Indeed, 
if  it  is  imagined  that  all  the  water  were  lifted  off,  the  centre  of  gravity 
of  the  Earth  would  still  be  at  the  centre  of  the  World.  . .  .  For,  essen 
tially,  the  earth  is  heavier  than  water.  Therefore,  whatever  may  be 
the  quantity  of  water  which  is  found  on  one  side  of  the  Earth  and  not 
on  the  other,  this  part  of  the  Earth  will  in  no  way  receive  more  help 
than  previously  in  counterpoising  and  pushing  awy  the  other  part.  ..." 

It  is  explained  "  that  one  part  of  the  Earth  rises  out  of  the  waters. 
The  Earth,  indeed,  is  not  uniformly  heavy,  so  that  its  centre  of  gravity 

1  DUHEM,  0.  $.,  Vol.  II,  p.  28. 


56  THE    ORIGINS 

is  placed  at  a  great  distance  from  its  geometrical  centre.  The  centre 
of  gravity  is  much  nearer  one  of  the  convex  hemispheres  that  define 
the  Earth  than  the  other.  Therefore  the  water,  which  is  unifirmly 
dense  and  tends  towards  the  centre  of  the  World,  runs  towards  that 
part  of  the  terrestrial  sphere  that  is  nearest  the  centre  of  gravity  of  the 
Earth,  so  that  the  other  hemisphere,  that  which  is  further  distant  from 
the  centre  of  gravity,  remains  uncovered.  " 

The  weakness  of  this  argument  is  clear.  But  undoubtedly  the 
theory  was  in  harmony  with  the  belief,  common  at  that  time,  in  the 
existence  of  a  terrestrial  hemisphere  completely  covered  by  a  vast 
ocean.  It  is  paradoxical  to  see  Albert  of  Saxony  thus  holding  that  the 
waters  of  the  sea  do  not  exert  any  heaviness,  but  this  is  in  accord  with 
a  more  general  thesis  that  is  indicated  below. 

Albert  of  Saxony  distinguished  between  heaviness  in  the  potential 
state,  that  of  a  heavy  body  occupying  its  natural  place,  and  the  actual 
heaviness  that  sets  a  body  in  motion  when  it  has  been  displaced  from  its 
natural  place  (or  shows  itself  as  a  resistance  to  obstacles  which  oppose 
the  body's  motion). 

We  shall  quote  from  Duhem's  commentary. 

"  The  parts  of  a  heavy  body,  be  they  solid  or  liquid,  do  not  push 
the  adjacent  parts  when  they  are  in  their  natural  place,  since  their 
heaviness  remains  in  its  potential  state.  Thus  the  bottom  of  the  sea 
does  not  support  any  load  or  any  pressure  that  is  due  to  the  water 
above  it.  In  all  circumstances  the  strength  of  the  heaviness,  whether  it 
is  habitual  or  actual*,  has  the  same  magnitude  in  the  same  heavy  body. 
A  part  of  the  earth  inclines  towards  its  natural  place  just  as  much  if  it 
is  placed  higher  up  than  if  it  is  lower  down.  " 

It  is  clear  that  this  thesis  contradicts  the  fundamental  axiom  of 
Jordanus — Gravius  esse  in  descendendo  quando  ejusdem  motus  ad  medium 
rectior.  Moreover,  it  is  not  surprising  that  Albert  of  Saxony  should  have 
rejected  the  idea  of  gravitas  secundum  situm,  and  have  substituted  for 
it  the  concept  of  a  greater  or  smaller  resistance  of  the  supporting  medium 
to  the  fall  of  a  moving  body. 


4.  ALBERT  OF  SAXONY'S  KINEMATICS.    THE  ACCELERATION  OF  FALUN*; 
BODIES. 

Whether  explicitly  or  not,  the  physicists  and  astronomers  of  Anti 
quity  treated  only  the  simple  uniform  motions  of  translation  and  rota 
tion,  and  confined  themselves  to  a  simple  qualitative  description  of 
accelerated  motion. 


THE    XlVth   CENTURY  57 

In  a  Xlllth  Century  manuscript1  there  is  the  statement  that  it  is 
correct  to  ascribe  the  velocity  of  its  mid-point  to  a  radius  turning  about 
its  centre.  This  text  is  mentioned  by  Thomas  Bradwardine,  proctor 
of  the  University  of  Oxford,  in  his  Tractatus  Proportionum  (1328). 
Bradwardine  denies  this  statement  and  attributes  the  velocity  of  its 
most  rapidly  moving  point  to  a  body  in  uniform  rotation. 

Albert  of  Saxony  stated  these  two  opinions  and  supported  that  of 
Bradwardine.  To  set  against  this,  he  gave  a  correct  definition  of  the 
angular  velocity  of  rotation  (velocitas  circuitionis) .  Further,  he  distin 
guished  between  deformed  motions,  in  which  the  velocity  of  a  moving 
body  varies  from  one  point  to  another,  and  irregular  motions,  in  which 
the  velocity  varies  from  one  instant  to  another. 

In  Book  II,  paragraph  XIII  of  his  Quaestiones?  Albert  of  Saxony 
examined  two  possible  laws  which  might  govern  the  fall  of  bodies — an 
increase  of  velocity  which  is  proportional  to  the  distance  travelled  and 
an  increase  proportional  to  the  time  taken. 

In  another  place,  he  rejects  these  two  laws,  which  lead  to  velocities 
which  become  infinite  with  the  distance  travelled  or  the  time  taken, 
and  contemplates  a  law  which  would  necessitate  that  the  velocity 
approach  a  finite  limit  when  the  time  increases  indefinitely.  On  this 
occasion,  Albert  of  Saxony  declares  himself  a  supporter  of  the  doctrine 
of  impetus  in  order  to  explain  the  acceleration  of  falling  bodies.  He 
observes,  however,  that  the  resistance,  increasing  more  quickly  than 
the  impetus  is  acquired,  will  limit  the  velocity  of  the  moving  body.  It  is 
important  to  notice  this  connection  between  Albert  of  Saxony  and 
John  Buridan's  doctrine,  and  to  recognise  the  considerable  authority 
which  the  latter's  work  had  over  this  long  period. 

5.  THE  DISCUSSION  OF  ACTION  AT  A  DISTANCE. 

In  commenting  on  Aristotle,  AverroSs  and  Albertus  Magnus  had 
held  that  the  weight  of  a  heavy  body  did  not  vary  with  its  distance 
from  the  centre  of  the  World.  On  the  other  hand  Saint  Thomas  Aquinas, 
arguing  from  the  acceleration  of  falling  bodies,  assumed  that  a  heavy 
body  increased  in  weight  as  it  approached  this  centre. 

In  the  XlVth  Century  John  of  Jandun,  in  his  commentary  on 
De  Caelo  (Book  IV,  para.  XIX),  declared  himself  for  the  first  opinion. 
The  natural  place  cannot  be  the  "  motor  "  of  a  heavy  body,  because  the 
motor  must  always  accompany  the  moving  body.  It  is  not  possible  to 
have  action  at  a  distance.  The  attraction  of  iron  by  a  magnet  presumes 

1  Biblioth&que  National^  Paris,  latin  collection,  Ms.  8680  A. 


58  THE    ORIGINS 

the  alteration  of  the  medium,  the  propagation  of  a  species  magnetica. 

On  the  other  hand,  William  of  Oekham  denied  that  the  motive  agency 
should  always  accompany  the  moving  body.  He  declared  that  iron  is 
attracted  at  a  distance  by  a  magnet  without  the  intermediary  of  any 
quality,  either  in  the  medium  or  the  iron.  He  assumed  that  the  magnet 
was,  in  itself,  the  total  cause  of  the  effect. 

In  his  Quaestiones  (Book  III,  para.  VII),  Albert  of  Saxony  held  that 
the  effect  of  a  body's  natural  place  on  a  body  was  different  from  the 
action  of  a  magnet  on  iron.  It  is  true  that  a  heavy  body  accelerates  in 
falling,,  "  but  its  initial1  velocity  is  not  greater  when  it  is  close  to  its  natural 
place  than  when  it  is  separated  from  it.  "  Thus  the  gravity  doeks  not 
depend  on  the  distance  from  the  centre  of  the  World — the  attraction  of 
a  magnet,  on  the  other  hand,  vanishes  at  some  distance. 

The  Schoolmen  of  the  XlVth  Century  therefore  rejected  the  hypo 
thesis  that  ascribed  weight  to  an  attraction  at  a  distance  exerted  by 
the  centre  of  the  Earth.  But,  as  Duhem  has  observed,  "  in  order  to 
prove  this  hypothesis  wrong,  it  was  necessary  to  work  out  its  conse 
quences.  They  [the  Schoolmen]  discovered  that,  on  the  basis  of  this 
supposition,  the  weight  of  a  body  would  vary  with  its  distance  from  the 
centre  of  attraction.  From  this,  it  was  argued  that  the  body  would 
have,  in  falling,  an  initial  velocity1  which  was  less  if  its  starting-point 
were  further  away  from  the  centre. "  2 

These  discussions  on  the  attraction  and  even  the  more  metaphysical 
argument  about  the  plurality  of  worlds  made  the  copernican  revolution, 
to  a  certain  extent,  possible. 


6.  NICOLE  ORESME — A  DISCIPLE  OF  BURIDAN. 

From  1348,  Nicole  Oresme,  of  the  diocese  of  Bayeux,  was  a  student 
of  theology.  In  1362  he  was  grand  master  of  the  College  of  Navarre. 
He  became  Bishop  of  Lisieux  on  August  3rd,  1377  and  died  there  on 
July  llth,  1382. 

Charles  V  entrusted  Oresme  with  the  task  of  translating  (into  French) 
and  annotating  certain  of  Aristotle's  works  which  had  previously  only 
been  accessible  to  the  scholars.  The  four  books  of  On  the  Heavens  and 
the  World  were  included  in  the  commission,  though  this  particular  part 

1  The  question  here  is  one  of  a  free  fall — the  initial  velocity  in  the  modern  Berne  in 
therefore  zero.  The  velocity  which  ALBERT  OF  SAXONY  intended,  however,  is  the  velocity 
acquired  after  a  very  short  time  if  the  body  starts  from  rest.     This  velocity  is  propor 
tional  to  the  weight  and  can  serve  as  a  measure  of  the  gravity. 

2  DUHEM,  JStudes  sur  Leonard  de  Vinci,  Series  II,  p.  90. 


THE    XlVth   CENTURY  59 

of  the  translation  was  never  printed.  At  the  beginning  of  the  XVIth 
Century  the  remainder  of  the  translation  (Ethics,  Politics  and  Economics) 
was  published,  together  with  Oresme's  Treatise  on  the  Sphere. 

In  dynamics,  Oresme  was  a  disciple  of  Buridan  and  adopted  from 
him  the  doctrine  of  impetus.  Thus  he  maintains,  in  his  Treatise  on  the 
Heavens  and  the  World,1  written  about  1377,  that  the  acceleration  of 
falling  bodies  is  not,  strictly  speaking,  accompanied  by  an  increase  of 
the  heaviness  of  the  body.  Rather  there  is  an  increase  of  an  "  accidental 
property  which  is  caused  by  the  reinforcement  of  the  isnelte  (velocity) 
and  this  property  can  be  called  impetuosity  (impetus).  " 

This  property  is  not  the  same  as  the  heaviness  "  because  if  a  hole 
were  to  be  dug  to  the  centre  of  the  earth  and  then  out  the  other  side, 
and  a  heavy  thing  were  to  fall  in  this  hole,  when  it  came  to  the  centre 
it  would  pass  beyond  it  and  rise,  through  the  agency  of  this  acquired 
and  accidental  property.  Then  it  would  fall  back  and  go  and  come  in 
the  way  that  we  see  in  a  heavy  thing  hanging  by  a  long  string.  There 
fore  this  is  not  strictly  heaviness,  since  it  is  able  to  make  [the  body] 
ascend.  " 

7.  ORESME'S  RULE  IN  KINEMATICS.    (UNIFORMLY  ACCELERATED  MOTION.) 

Oresme  was  above  all  a  mathematician,  and  in  this  capacity  he 
emerges  as  Descartes'  forerunner  in  the  matter  of  the  invention  of 
co-ordinates.  As  will  be  seen,  and  as  Moritz  Cantor  has  pointed  out,2 
we  shall  not  stray  from,  the  subject  in  hand  if  we  emphasise  this  aspect 
of  his  work. 

We  shall  follow  the  Tractatus  de  Jiguratione  potentiarurn  et  mensu- 
rarum  difformitatum^  Oresme  starts  from  the  principle  that  every 
measurable  thing  can  be  thought  of  as  a  continuous  quantity.  Each 
intensity  can  be  represented  by  means  of  a  straight  line  erected  vertical 
from  each  point  of  the  "  subject  "  which  affects  the  intensity. 

Extension  (longitudo)  ia  represented  diagrammatically  by  a  horizon 
tal  line  drawn  in  the  direction  of  the  subject.  At  each  point  of  this  line 
a  vertical  is  erected  whose  height  (altitudo  or  latitudo)  is  proportional 
to  the  intensity  (intensio)  of  the  property  at  the  point  corresponding  to 
the  subject. 

Thus  the  triangle  of  figure  16  represents  a  uniformly  deformed 
quality  (uniformiter  difformis)  terminated  at  a  value  zero.  The 

1  Bibliothtque  National^  Paris,  frcnch  collection,  Ms,  1083. 

2  Vorlesungen  uber  die  Ceschichte  der  Math^matik,  2nd  Ed.,  Vol.  II,  p,  129.  (Leipzig, 
1900.) 

8  Bibliothdque  National^  Paris,  latin  collection,  Ms.  7371,  Trans.  DUHEM. 


60  THE    ORIGINS 


a)  b)  c)  d) 

Fig.  16 

rectangle  6  represents  a  uniform  quality  and  the  trapezium  c  a  uniform- 
ely  deformed  quality  terminated  by  certain  values  at  one  end  and  at 
the  other.  Any  other  quality  is  said  to  be  deformably  deformed — that 
is,  non-uniformly  deformed  or  non-uniformly  varying  (difformiter 
difformis) .  Such  a  one  can  be  represented  in  the  same  way  by  erecting 
a  vertical  proportional  to  the  intensity  from  each  point  of  the  extension. 
Oresme  pointed  out  explicitly  that  the  scale  of  such  a  representation 
could  be  chosen  at  will.  Therefore  the  same  quality  can  be  represented 
by  diagrams  whose  verticals  are  in  a  given  relation  to  each  other.  Thus 
Oresme  understands  that  the  same  quality  can  be  represented  by  a 
diagram  which  is,  for  example,  either  circular  or  elliptical.  He  then 
proceeds  to  a  classification  of  deformities  according  to  the  direction  of 
their  concavity  and  according  to  whether  they  arc  rational  (circular) 
or  not.  In  this  way  he  was  able  to  enumerate  62  different  kinds  of 
deformity. 

Oresme  came  very  near  to  modern  analytical  geometry  when  he 
wrote,  "  A  uniformly  deformed  quality  is  such  that  when  any  three 
points  of  the  subject  are  given,  the  relation  of  the  interval  between  the 
first  and  the  second  to  the  interval  between  the  second  and  the  third 
is  the  same  as  the  relation  of  the  excess  of  intensity  of  the  first  over  the 
second  to  the  excess  intensity  of  the  second  over  the  third.  "  This 
statement  expresses  the  relation  between  the  co-ordinates  of  three  points 
on  a  straight  line  explicitly. 

Oresme  went  further  than  this  in  envisaging  superficial  qualities 
— qualities  which  had  two  dimensions  with  respect  to  the  subject  and 
whose  intensity  must  be  represented  by  a  normal  to  a  plane  surface 
which  defines  the  extension.  Similarly,  he  put  the  question  of  how  a 
corporeal  quality — one  having  three  dimensions  with  respect  to  the  sub 
ject — can  be  represented.  This  passage  merits  quotation. 

"  A  superficial  quality  is  represented  by  a  solid  figure.  Now  a 
fourth  dimension  does  not  exist  and  it  is  impossible  to  conceive  of  one. 
Nevertheless  a  corporeal  quality  may  be  thought  of  as  having  a  double 
corporeity.  One  in  a  real  extension,  through  the  effect  of  the  extension 
of  the  subject,  has  a  locus  in  all  dimensions.  But  there  is  also  another 
which  is  only  imagined  and  which  arises  from  the  intensity  of  the  qua- 


THE    XlVth    CENTURY 


61 


lity.     This  quality  is  repeated  an  infinite  number  of  times  by  the  multi 
tude  of  surfaces  which  may  be  traced  with  respect  to  the  subject.  " 

In  kinematics,  Oresme  accepted  Albert  of  Saxony's  ideas  but  ex 
pressed  them  with  the  help  of  his  graphical  representation.  Velocity 
is  susceptible  of  a  double  extension,  either  in  time  or  with  respect  to 
the  subject.  It  can  be  uniform  or  deformed  with  respect  to  each  of 
these  two  extensions. 

Further,  Oresme  defined  the  total  quality  or  the  measure  of  a  quality 
which  was  linear  (or  superficial)  with  respect  to  the  subject  as  the  area 
(or  the  volume)  of  the  diagram  which  represents  it. 

It  is  clear  that  if  time  is  taken  as  the  variable  of  extension,  the 
measure  or  total  quality  of  the  velocity  of  a  uniform  motion  is  equal 
to  the  distance  travelled.  Oresme 

did  not  confine    himself    to   this  4i 

instance.  He  contemplated  a  suc 
cession  of  uniform  motions  in  the 
following  way. 

He  divides  the  time  t  into 
proportional  parts  which  form  a 
geometrical  progression  with  ratio 

—  and  the  first  term  -.     The  velo- 

2.  .  2. 

city   has   intensity   ni   in  the  nth 

interval.     Under  these  conditions  Oresme  states  that  the  total  distance 

travelled  is   equal  to  four  times  the  first  rectangle,  that  is  4   UH  • 

Oresme  stated  the  following  general  rule  for  uniformly  deformed  qua 
lities —  "  Omnis  qualitas,  sifuerit  umformiter  difformis,  secundum  gradum 
puncti  medii  ipsa  cst  tanta  quanta  qualitas  cjusdem  subjecti.  "  That  is, 
any  uniformly  deformed  quality  lias  the  same  total  quality  (meas 
ure)  as  if  it  were  related  to  the 
subject  with  the  value  which  it 
takes  at  the  middle  point. 

Oresme  verifies  this  rule  for  a 
linear  uniformly  deformed  quality 
that  starts  with  an  intensity  AC 
at  A  and  finishes  with  zero  value 
at  J3.  If  D  is  the  centre  of  the 
line  AB  which  represents  the 


21 

Si 

1 

t                t       t:   t 
2               4       6  16 

Fig.  17 


° 
Fig.  18 


B 


1  It  should  be  remarked  in  passing  that  this  fact  shows  that  Oresme  knew  how  to 
calculate  the  sum  of  the  series  whose  general  term  is—  • 


62  THE    ORIGINS 

subject  (subjectiva  linea),  the  corresponding  intensity  is  DE.  The 
uniform  quality  that  has  the  rectangle  AFGB  as  its  measure  has  the 
same  measure  as  the  uniformly  deformed  quality  represented  by  the 
triangle  ACB,  because  the  area  ACE  is  equal  to  the  area  AFGB. 
Oresme  declares,  "Any  uniformly  deformed  quality  or  velocity 
is  thus  found  to  be  equivalent  to  a  uniform  velocity,  "  but  he  does  not 
explain,  at  this  point,  the  identity  of  the  measure  and  the  distance 
travelled.  So  that  he  does  not  apply  the  rule  which  he  has  just  for 
mulated  to  a  uniformly  deformed  motion,  although  this  rule  includes 
the  law  of  distance  travelled  in  such  a  motion.  Indeed,  in  modern  lan 
guage  we  may  write  simply 


We  have  seen  that  some  of  the  Schoolmen  discussed  the  fall  of 
bodies  while  others  were  concerned  with  kinematics,  and  that  each  of 
these  sections  developed  a  representation  which  contained  a  key  to 
the  law  of  the  distance  travelled  by  a  moving  body.  But  the  union 
of  these  two  problems  was  not  effected.  Undoubtedly  the  reason  for 
this  lies  in  the  fact  that  the  Schoolmen  were  satisfied  when  they  had 
constructed  abstract  systems,  whose  niceties  distracted  their  attention 
from  the  rudimentary  experimental  basis  which  they  possessed. 

8.  ORESME  AS  A  PREDECESSOR  OF  COPERNICUS. 

Except  to  the  extent  that  it  impinges  on  the  history  of  the  prin 
ciples  of  mechanics,  we  are  not  here  concerned  with  the  history  of 
world-  systems.  However,  by  considering  this  aspect  of  Orcsme's  nrnst 
original  work,  we  shall  see  him  to  have  been  a  prophet  of  Copernicus.1 

The  following  quotations  are  taken  from  Nicole  Oresrne'B  Treatise 
on  the  Heavens  and  the  World. 

Aristotle  had  established,  in  the  second  book  of  De  Caelo?  that  the 
Earth  remained  motionless  at  the  centre  of  the  World*  Grew  me 
declared,  "  no  observation  could  prove  that  the  Heavens  moved  with 
a  diurnal  2  motion,  and  that  the  Earth  did  not.  " 

In  this  connection  he  made  use,  in  an  especially  complete  way,  of 
the  relativity  of  all  motion. 

"  If  a  man  were  placed  in  the  Heavens,  suppose  that  he  were  moved 
with  a  diurnal  motion.  Then  if  this  man  who  is  carried  above  the 
Earth  sees  the  Earth  clearly,  and  picks  out  the  mountains,  the  valleys, 

1  Cf.  DUHEM,  Revue  gfafrale  des  Sciences,  Nov.  15,  1909. 

2  Lit.,  "journal." 


THE    XlVth   CENTURY  63 

rivers,  towns  and  castles,  it  will  seem  to  him  that  that  the  Earth  is 
moved  diurnally  just  as,  to  us  on  the  Earth,  the  Heavens  seem  to 
move.  And  similarily,  if  the  Earth  is  moved  with  a  diurnal  motion 
and  the  Heavens  not,  it  will  seem  to  us  that  the  Earth  is  still  and  that 
the  Heavens  move.  " 

Oresme  discussed  Aristotle's  argument  according  to  which  a  stone 
thrown  vertically  upwards  should  fall  to  the  west  if  the  Earth  is  not 
at  rest.  In  this  connection,  he  declares  that  a  stone  thrown  vertically 
in  this  way  would  be  carried  very  rapidly  towards  the  east,  "  together 
with  the  air  through  which  it  passes  and  with  all  the  mass  of  the  lower 
part  of  the  World  "  which  participates  in  the  diurnal  motion.  In 
short,  Oresme  believed  that  the  stone  links  its  motion  with  that  oft 
the  Earth,  from  which  it  originally  obtained  its  impetus.  This  thesis 
is  correct,  at  least  in  its  essentials.1 

In  a  similar  way,  Oresme  gave  the  lie  to  Ptolemy's  argument  that 
an  arrow  shot  vertically  from  the  deck  of  a  ship  moving  very  rapidly 
towards  the  east  will  fall  far  to  the  west  of  the  ship.  He  then  discussed 
the  following  reasons  which  had  been  given  in  support  of  the  hypothesis 
that  the  Heavens  moved  and  the  Earth  was  stationary. 

1)  Any  simple  body  can  only  have  a  simple  motion.     The  Earth 
can  only  have  a  natural  falling  motion. 

2)  Apart  from  this  natural  falling  motion  could  not  have  a  circular 
motion — this  motion,  "  which  is  violent,  could  not  be  perpetual.  " 

3)  AverroSs  holds  that  any  motion  from  place  to  place  can  be 
related  to  a  body  at  rest  and,  for  this  reason,  he  assumes  that  the 
Earth  is  necessarily  fixed,  at  the  centre  of  the  Heavens. 

4)  All   motion    supposes    a   "  motive    virtue. "     Now    the    Earth 
cannot  be  moved  circularly  by  means  of  its  heaviness.     "  And  if  it  is 
moved  in  this  way  by  some  outside  agency,  such  a  motion  will  be 
violent   and  not   perpetual.  " 

5)  "  If  the  Heavens  were  not  moved  with  a  diurnal  motion,  all 
Astrology  2  would  be  false.  " 

1  We  now  know  that,  in  a  free  downward  fall,  a  heavy  moving  body  HufFers  a  small 
deflection  towards  the  East  as  a  result  of  the  rotation  of  the  Earth.     The  complete 
calculation  requires  that  account  he  taken  of  the  compound  centrifugal  forcet  but  a 
very  simple  intuitive  argument  can  give  the  direction  of  this  deflection,  by  means  of 
the  hypothesis  that  the  motion  of  the  body,  starting  from  rest,  proceeds  according  to 
the  law  of  areas  ^  ^  c<mgtant 

Initially,  r  ===  r0  and  6  =  co,  the  velocity  of  rotation  of  the  Earth.  During. the  motion  r 
will  decrease.  The  inequality  r  <  r<>  requires,  by  the  law  of  areas,  that  0  >  to,  which 
shows  that  the  body  is  diverted  towards  the  East. 

2  Read  "  Astronomy.  " 


64  THE    ORIGINS 

6)  Motion  of  the  Earth  would  contradict  the  Holy  Scriptures-- 
44  Oritur  sol  et  occidit,  et  ad  locum  suum  revertitur  .  .  .  Deus  firmavit 
orbem   Terrae   qui   non  commovebitur.  " 

7)  The  Scriptures  also  say  that  the  Sun  stopped  in  Joshua's  time, 
and  that  is  started  its  journey  again  in  the  time  of  King  Hezekiah. 
And  if  it  was  the  Earth  which  moved,  not  the  Heavens,  "  such  a  cess 
ation  would  have  been  in  versed.  " 

Oresme's  replies  to  these  arguments  were  the  following. 

1)  It  is  more  reasonable  to  believe  that  every  simple  body  and  part 
of  the  World,  except  for  the  Heavens,  is  activated  by  a  rotational 
motion  in  its  natural  place.     And  that,  if  a  part  of  such  a  body  is  dis 
placed  from  its  place  in  the  whole  it  will,  if  this  is  allowed,  return  there 
as  directly  as  possible. 

2)  The  rotational  motion  of  the  Earth  is  certainly  a  natural  one,  but 
parts  of  the  Earth  that  are  displaced  from  their  accustomed  position 
have   a  natural,   ascending  or  descending,  motion. 

3)  "  Supposing  that  circular  motion  requires  the  presence  of  some 
other  body  at  rest,  "  it  does  not  follow  that  the  body  at  rest  should 
be  inside  the  body  which  moves,  for  there  is  nothing  at  rest  inside  a 
grindstone  "except  a  single  mathematical  point,  which  is  not  a  body." 

4)  The  virtue  which  produces  the  rotational  motion  of  the  lower  part, 
of  the  World  is  the  nature  of  this  part.     It  is  the  same  thing  that  also  pro 
duces  the  motion  of  the  Earth  towards  its  natural  place  when  it  has  been 
displaced  from  it,  in  the  same  way  that  iron  is  drawn  towards  a  magnet. 

5)  All  appearances,  all  conjunctions,  oppositions,  constellations  and 
influences   in  the   Heavens  remain   unchanged  when   it   is   supposed 
that  the  motion  of  the  Heavens  is  apparent  and  the  motion  of  the 
Earth  is  real. 

6)  The  Holy  Scriptures  are  consistent  here,  "  in  the  manner  of 
ordinary  human  speech,  "  to  the  same  extent  that  they  agree  in  many 
other  places.     Things  are  not  "  as  the  letter  sounds.  "     Thus  it  is 
written  that  God  covered  the  Heavens  with  clouds —  "  Qui  opcrit  ("aelttm 
nubibus.  "     Now,  on  the  contrary,  all  the  evidence  shows  that  it  LH 
the  Heavens  which  cover  the  clouds.     Here  again  the  words  indicate 
the  appearance  and  not  the  truth.     It  is  the  same  for  the  motion  of 
the  Earth  and  the  Heavens. 

7)  The  stopping  of  the  Heavens  in  the   time  of  Joshua  was   an 
illusion — in  fact,  it  was  the  Earth  which  stopped,  and  which  started 
its  motion  again,  or  accelerated,  in  Hezekiah's  time. 

Oresme  then  gives  us  several  "  good  reasons  "  intended  to  show 
that  the  Earth  has  a  diurnal  motion  and  the  Heavens  do  not. 


THE    XlVth    CENTURY  65 

Every  thing  which  benefits  from  another  thing  must  set  itself 
to  receive,  by  its  own  motion,  the  benefit  which  it  obtains  from  the 
other.  Thus  each  element  is  moved  towards  its  natural  place,  where 
it  is  kept.  On  the  other  hand,  the  natural  place  does  not  move  towards 
the  element. 

From  which  it  follows  that  the  Earth  and  the  elements  on  the 
Earth,  which  benefit  from  the  heat  and  the  influence  of  the  Heavens, 
must  arrange  themselves  by  their  own  motion  to  duly  receive  this 
benefit ;  "  also,  to  speak  familiarly,  as  something  which  is  roasted 
at  the  fire  receives  the  heat  of  the  fire  about  itself  because  it  is  turned, 
and  not  because  the  fire  is  turned  towards  it.  " 

It  is  natural  that  the  motions  of  the  simple  bodies  of  the  World 
should  have  the  same  direction.  Now,  according  to  the  Astronomers, 
it  is  impossible  that  all  these  motions  should  take  place  from  east  to 
west.  On  the  contrary,  if  it  is  assumed  that  the  Earth  moves  from  west 
to  east  then  this  will  agree  with  the  other  motions,  "  the  Moon  in  one 
month,  the  Sun  in  a  year,  Mars  in  about  two  years  and  similarly  for 
the  others.  " 

In  this  way,  the  part  of  the  Earth  which  is  habitable  will  be  at  the 
top  and  on  the  left  of  the  World.  u  And  it  is  reasonable  that  human 
habitation  should  be  found  in  the  most  noble  place  that  there  is  in  the 
Earth.  " 

According  to  Aristotle,  the  most  noble  thing  which  is  and  can  be 
has  its  perfection  at  rest.  Terrestrial  bodies  are  set  in  motion  towards 
their  natural  place  in  order  to  rest  there.  We  pray  that  God  should 
give  the  dead  rest — Requiem  aetcrnam.  .  .  .  The  Earth,  the  most  com 
mon  thing,  is  displaced  more  rapidly  than  air,  the  Moon  or  the  stars. 

In  the  hypothesis  of  the  stationary  Earth,  the  velocities  which  must 
be  assigned  to  the  stars,  because  of  their  distance  from  the  centre,  are 
inadmissible. 

The  constellation  of  the  North  Wind— Major  Ursa— -does  not  turn 
round,  with  the  chariot  in  front  of  the  cattle,  as  it  would  if  it  partook 
of  a  diurnal  motion. 

All  the  appearances  can  be  saved  by  means  of  a  minor  change — the 
diurnal  motion  of  the  Earth,  whose  size  is  so  small  in  comparison  with 
the  Heavens — and  without  demanding  so  many  different  and  incredible 
processes  that  God  and  Nature  would  have  created  for  no  purpose. 
By  this  means,  the  introduction  of  a  IXth  sphere  is  also  made  unneces 
sary. 

When  God  accomplishes  a  miracle,  he  does  so  cc  without  changing 
the  common  course  of  nature,  except  to  the  extent  that  this  must  be 
done.  '*  Thus  it  is  natural  that  the  arrest  of  the  Sun  in  Joshua's  time, 


66  THE    ORIGINS 

and  the  start  again  in  Hezekiah's  time,  should  be  the  result  of  terrestrial 
motion  alone. 

Oresme  concluded  that  considerations  such  as  these  "  are  valuable 
for  the  defence  of  our  Faith.  "  More  astute  than  Galileo,  and  safe 
from  the  thunderbolts  that  were  to  be  hurled  at  this  thesis  later,  he 
was  nominated  Bishop  of  Lisieux  in  reward  for  his  work. 

Since,  as  we  have  said,  Oresme's  Treatise  on  the  Heavens  and  the 
World  was  never  printed,  it  is  very  unlikely  that  his  ideas  on  the  diurnal 
motion  of  the  Earth  could  have  become  available  to  Copernicus. 

Perhaps  the  reader  will  decide  that  we  have  devoted  too  much 
attention  to  this  early  philosopher.  But  we  have  seen  the  best  and  the 
worst  of  Oresme's  arguments  about  the  system  of  the  World.  We  have 
felt  the  mood  of  the  time,  at  once  naive  and  acute,  fantastic  and  serious, 
familiar  and  dogmatic.  Of  the  originality  of  Oresme  as  a  mathemati 
cian,  and  of  the  vigour  of  his  penetrating  thought,  there  is  no  doubt. 
The  prejudices  of  the  Schools  and  the  accepted  ideas  of  the  time  did  not 
imprison  him.  In  the  field  of  mechanics  he  was  one  of  the  first  to 
address  himself  to  the  great  French  public,  or,  as  he  said  himself,  more 
accurately,  "  to  all  men  of  free  condition  and  noble  intellect.  " 


9.  THE  OXFORD  SCHOOL. 

In  his  study  of  the  representation  of  qualities  Oresme  invoked  the 
authority  of  certain  veteres  whom  he  did  not  name.  It  is  reasonable 
to  believe  that  the  ancients  who  preceeded  Oresme  in  the  general  study 
of  forms  were  the  logicians  of  the  Oxford  School. 

One  of  the  most  eminent  masters  of  this  school  was  William  Hey  tes- 
bury,  or  Hentisberus.  It  is  said  that  Heytesbury  was  a  fellow  of 
Merton  College  in  1330,  that  he  belonged  to  Queen's  College  about  1340 
and  that  he  was  Chancellor  of  the  University  of  Oxford  in  1371, 

Primarily  Heytesbury  was  a  logician  of  the  most  acute  kind.  But 
he  was  also  concerned  with  kinematics,  and  it  is  in  this  connection  that 
he  claims  our  attention.  In  his  Regulae  solvendi  sophismata  the  follow 
ing  rule  is  given  without  proof—  when  the  velocity  of  a  moving  body 
increases  with  the  time  in  such  a  way  that  it  is  uniformly  deformed, 
in  a  given  time  the  body  travels  the  same  path  as  if  it  had  moved  uni 
formly  with  the  velocity  acquired  half  way  through  this  time.  This 
is  Oresme's  rule  applied  specifically  to  distance. 

To  set  against  this,  Heytesbury  supported  Thomas  Bradwardme*B 
opinion— referred  to  earlier— that  the  effective  velocity  of  a  rotating 
body  was  that  of  its  most  rapidly-moving  point. 


THE    XlVth   CENTURY  67 

The  most  remarkable  feature  of  Heytesbury's  work  is  the  appearance, 
albeit  shrouded  in  obscurity,  of  the  concept  of  acceleration.  This  was 
unknown  to  the  Paris  School. 

In  fact,  in  his  treatise  De  Tribus  praedicamentis,  Heytesbury  distin 
guished  between  the  latitudo  motus  (velocity)  and  the  velocitas  intensionis 
vel  remissionis  motus  whose  value  was  the  increase  or  the  decrease  of  the 
former.  This  quantity  corresponds  to  acceleration. 

"  For  a  moving  body  which  starts  from  rest  there  can  be  imagined 
a  range  of  velocity  which  increases  indefinitely.  In  the  same  way  can 
be  imagined  a  range  of  acceleration  or  of  slowing  down  (latitudo  inten 
sionis  vel  remissionis)  according  to  which  a  body  can  accelerate  or  slow 
down  its  motion  with  an  infinitely  variable  quickness  or  slowness. 
This  second  range  is  related  to  the  range  of  motion  (velocity)  as  the 
motion  (velocity)  is  related  to  the  magnitude  (distance)  that  may  be 
travelled  in  a  continous  manner.  "  1 

Through  this  obscure  language  we  catch  the  first  glimpse  of  quantities 
that  have  become  familiar  tools  of  our  trade,  the  vector  representing 

J  O 

the  distance  travelled,  S ;  the  velocity  (vector  derivative),  —  ;   and  the 

d*S 

acceleration  (vector  derivative  of  the  velocity),  ~-r%- 

at 

We  must  also  refer  to  the  Liber  Calculationum.  In  order  to  avoid 
the  labyrinths  of  the  Oxford  School,  we  shall  confine  ourselves  to  a 
mention  of  Swineshead  (variously  called  Suincet,  Suisset,  Suisseth, .  . . 
by  the  continental  copyists).  The  tradition  of  the  XVth  and  XVIth 
Centuries,  on  the  publication  of  Liber  Calculationum  in  1488,  1498  and 
1520,  added  the  epithet  "  Calculator  "  to  this  name.  The  document  is 
the  most  typical  of  the  Oxford  dialectic  that  is  available  to  us,  and  in 
spite  of  the  relentless  attacks  of  the  Humanists,  it  was  very  highly 
regarded  until  the  XVII  th  Century.  Thus  Leibniz,  writing  to  Wallis, 
could  express  his  wish  to  see  it  republished. 

Unfortunately,  this  work  has  only  been  attributed  to  Swineshead  in 
error.  Duhem,  that  tireless  investigator,  found  a  manuscript2  of  this 
work  which  goes  back  to  the  XlVth  Century,  and  in  which  the  copyist 
attributes  the  work  to  Ricardus  of  Ghlymi  Eshedi.  (This  must  refer 
to  William  Collingham,  a  Master  of  Arts  of  Oxford.) 

This  treatise  is  concerned  with  the  general  theory  of  forms,  and 
sophist  discussions  make  up  the  essential  part  of  it.  We  shall  quote  a 
single  extract  from  Chapter  XV,  which  is  called  De  medio  uniformiter 
difformi. 

I  Venice  Edition,  1494. 

II  Biblioth&que  Nationals,  Paris,  latin  collection,  Ms.  6558, 


68  THE    ORIGINS 

"  If  the  motion  of  a  body  is  uniformly  accelerated  and  starts  with 
a  value  zero,  the  body  will  travel  three  times  further  in  the  second  half 
of  the  time  than  in  the  first  half.  " 

This  is  a  direct  corollary  of  the  law  of  distances  in  uniformly  varying 
motion. 

In  this  way  then,  in  XlVth  Century  Oxford,  the  kinematics  of  uni 
formly  varying  motion  was  known  and  commonly  taught.  The  English 
School  has  the  merit  of  having  stated  the  law  of  distances  more  precisely 
than  the  School  at  Paris — on  the  other  hand,  it  seems  to  have  neglected 
Oresme's  remarkable  representation  of  uniformly  deformed  qualities. 

Just  as  much  at  Oxford  as  at  Paris,  these  developments  in  kinematics, 
and  the  related  general  study  of  properties  in  intensity  and  extension, 
had  no  influence  on  the  study  of  the  fall  of  bodies.  The  description  of 
these  phenomena  remained  completely  qualitative. 

10.    THE  TRADITION  OF  ALBERT  OF   SAXONY  AND   OF  BlJRIDAN. 

The  tradition  of  Albert  of  Saxony  and  of  Buridan  was  preserved  in 
France  and  Germany  by  Themon,  a  son  of  Jew  who  taught  at  Paris  in 
1350,  and  Marsile  of  Inghen,  who  became  rector  of  Heidelberg  in  1386 
after  having  been  at  the  University  of  Paris  in  1379.  It  is  noteworthy 
that  Marsile  modified  Buridan's  doctrine  in  a  somewhat  unfortunate 
way.  Thus  he  held  that  the  impetus  was  at  first  weak  in  those  parts 
of  a  body  that  were  not  in  contact  with  the  motive  agency,  and  that  it 
was  strengthened  there  as  the  whole  impetus  became  uniformly  distri 
buted  throughout  the  moving  body. 

We  must  also  refer  to  Pierre  d'Ailly  (1330-1420)  who  was  high  master 
of  the  College  of  Navarre  in  1384  and  who  added  the  following  original 
items  to  Albert  of  Saxony's  paradoxes. 

"  Someone  who  owns  a  field  adjoining  another  piece  of  land,  mid 
who  excavates  his  earth  in  such  a  way  that  the  area  of  the  cavity  remains 
constant,  is  defrauding  his  neighbour. 

"  If  the  Earth  is  cut  by  a  plane  surface  whose  centre  is  at  the  centre 
of  the  World,  when  water  is  poured  on  this  plane  it  will  tend  to  assume 
the  form  of  a  hemisphere. 

"  In  the  second  place,  if  the  bottom  of  a  pool  is  flat,  this  pool  will 
certainly  be  deeper  in  the  middle  than  at  the  sides.  ..." 

Pierre  d'Ailly  also  gave  Roger  Bacon's  paradox  which  has  been 
quoted  above  (p.  52). 

With  such  intellectual  games  did  the  Schoolmen  of  the  XlVth  Cen 
tury  delight  themselves.  So  alive  was  this  tradition  that  it  maintained 
itself  for  over  two  hundred  years. 


CHAPTER    FIVE 


XVth  AND  XVIth  CENTURIES 

THE  ITALIAN  SCHOOL 

BLASIUS  OF  PARMA        THE  OXFORD  TRADITION 
NICHOLAS  OF  CUES  AND  LEONARDO  DA  VINCI 

NICHOLAS  COPERNICUS 
THE  ITALIAN  AND  PARISIAN  SCHOOLMEN 

OF  THE  XVIth  CENTURY 
DOMINIC  SOTO  AND  THE  FALL  OF  BODIES 


1.  BLASIUS  OF  PARMA  AND  HIS  TREATISE  ON  WEIGHTS. 

Blasius  of  Parma  (Biagio  Pelacani),  who  became  a  doctor  at  Padua 
in  1347,  taught  at  Padua  and  Bologna.  He  went  to  Paris  about  1405 
and  died  at  Parma  in  1416. 

His  Treatise  on  Weights  is  known  to  us  through  a  copy  made  by 
Arnold  of  Brussels  and  dated  1476. 

This  treatise  derives  from  Jordanus'  School  and  links  up  the  idea 
ofgravitas  secundum  situm — a  first  principle  of  Xlllth  Century  statics — 
with  the  tendency  of  a  heavy  body  to  fall  along  a  chord  rather  than 
along  an  arc  of  a  circle,  and  thus  to  take  the  shortest  path,  in  Aristotle's 
sense,  to  its  natural  place. 

Blasius  of  Parma  observed  that  when  a  balance  with  equal  arms 
supporting  equal  weights  is  moved  away  from  the  centre  of  the  World, 
these  weights  will  appear  to  become  heavier.  Indeed,  the  line  along 
which  each  of  the  weights  tends  to  fall  makes  an  angle  with  the  vertical 
through  the  point  of  support  which  is  the  more  acute  as  the  balance  is 
the  further  away  from  the  centre  of  the  World.  This  embellishment  adds 
no  thing  useful  to  the  positive  statics  of  the  authors  of  De  Ponderibus. 

In  a  general  way,  Blasius  of  Parma  was  a  critic  and  a  sceptic  who 
was  content  to  multiply  the  objections  to  his  predecessors'  theories.  For 
example,  he  observed  that  it  is  necessary  to  take  account  of  passive 
resistances,  though  the  correctness  of  the  propositions  of  statics  depends 


70  THE    ORIGINS 

on  the  process  of  neglecting  all  resistances  occasioned  by  the  medium. 
In  a  very  naive  way  Blasius  of  Parma  attempted  to  take  these  resistances 
into  consideration. 

Apart  from  his  Treatise  on  Weights,  Blasius  of  Parma  also  wrote  Quaes- 
tiones  super  tractatu  de  latitudinibus  formarum,  which  was  printed  in  1486. 
In  it  he  appears  as  an  unsophisticated  commentator  of  Oresme's  doctrine. 

Although  a  critic  of  no  great  originality,  Blasius  of  Parma  was  one 
of  the  means  by  which  the  statics  of  the  Xlllth  Century  and  the  kine 
matics  of  the  XlVth  Century  were  handed  on  to  the  Italian  School, 
which  was  destined  to  dominate  mechanics  during  the  period  that  we 
are  going  to  study. 

2.  THE  ITALIAN  TRADITION  OF  NICOLE  ORESME  AND  THE  OXFORD  SCHOOL. 

Together  with  Blasius  of  Parma,  we  must  refer  to  Ga^tan  of  Tiene. 
Like  Blasius,  this  author  taught  at  Padua  and  died  there  in  1465,  and 
he  is  responsible  for  having  preserved  the  tradition  of  William  Heytes- 
bury  and  the  "  Calculator  "  in  Italy.  One  by  one,  he  annotated  the 
sophisms  of  the  Oxford  School,  and  his  work  was  printed  in  Venice  in 
1494,  together  with  the  works  of  Heytesbury. 

In  particular,  GaStan  of  Tiene  emphasised  the  distinction  between 
latitude*  motus  (velocity)  and  latitude  intensionis  motus  (acceleration). 
In  this  way  the  Italian  School  explained,  more  clearly  than  Heytesbury 
had  done,  the  fact  that  a  uniformly  deformed  motion  corresponds  to  a 
constant  latitude  intensionis  motus — that  is,  to  a  constant  acceleration  ; 
and  that  a  deformably  deformed  motion  corresponds  to  a  uniformly 
deformed  latitude  intensionis  motus. 

Bernard  Torni,  a  Florentian  physician  who  died  about  1500,  carried 
on  the  work  of  Gaetan  of  Tiene,  and  published  Annotata  to  Heytesbury's 
treatise  which  made  frequent  mention  of  the  "  Calculator.  "  He  was 
equally  enthusiastic  about  Oresme's  analysis,  though  he  was  only  con 
cerned  with  the  arithmetical  procedures  contained  in  this  work. 

John  of  Forli,  who  taught  medicine  at  Padua  about  1409  and  died 
there  in  1414,  wrote  a  treatise  De  intensione  et  remissions  formarum 
which  was  printed  at  Venice  in  1496.  In  it  he  refuted  W.  Burley, 
rejected  Oresme's  rule  for  the  evaluation  of  a  uniformly  deformed  quality, 
and  attempted  to  introduce  into  medicine  a  terminology  which  was 
inspired  by  the  Oxford  School.  The  Humanists,  especially  VivSs,  made 
him  their  target. 

It  may  be  inferred,  as  Duhem  has    remarked,1  "  that   thanks  to 

1  DUHEM,  Etudes  sur  Leonard  de  Finci,  Series  III,  p.  509. 


XVth   AND    XVIth   CENTURIES  71 

Nicole  Oresme,  William  Heytesbury  and  the  *  Calculator  \  at  the 
middle  of  the  Quattrocento  the  Italian  masters  were  well- acquainted 
with  all  the  laws  of  uniformly  accelerated  or  uniformly  retarded  motion. 
But  it  seems  that  none  of  them  was  inspired  to  assume  that  the  fall  of 
bodies  was  uniformly  accelerated  or,  for  this  reason,  to  apply  these 
laws  to  that  phenomenon.  " 

3.  NICHOLAS  OF  CUES  (1404-1464)  AND  THE  DOCTRINE  OF  "  IMPETUS 
IMPRESSUS.  " 

Nicholas  of  Cues  studied  at  Heidelberg  from  1416,  and  later  at  Padua 
in  1424.  On  returning  to  Germany  he  devoted  himself  to  theology  and 
science.  He  became  Bishop  of  Brixen  (Tyrol)  in  1450  and  died  at  Todi 
(Umbria)  on  August  llth,  1464, 

His  works  were  published  in  three  parts  between  1500  and  1514, 
and  later  reprinted  at  Basle  in  1575. 

Nicholas  of  Cues  was  primarily  a  metaphysician.  In  De  docta 
ignorantia  he  maintained  that  it  was  impossible  to  accept  the  idea  of 
absolute  truth,  and  argued  the  identity  of  the  absolute  maximum  and 
the  absolute  minimum,  as  well  as  the  existence  of  an  Universe  at  once 
finite  and  unlimited. 

In  mechanics,  Nicholas  of  Cues  has  left  us  the  dialogues  De  ludo  globi. 
He  is  concerned  with  a  game  where  a  hemisphere  is  thrown  in  such  a 
way  that  it  meets  some  pins  which  are  arranged  in  a  spiral.  The  problem 
is  to  explain  the  trajectory  of  the  body.  Further,  in  the  dialogue  De 
Possest^1  he  concerned  himself  with  the  gyroscopic  motion  of  a  toy  top. 

"  A  child  takes  up  this  dead  toy,  devoid  of  motion,  and  wishes  to 
make  it  live.  For  this  purpose,  by  a  procedure  which  he  has  invented 
and  which  is  the  instrument  of  his  intelligence,  he  impresses  on  the  toy 
the  permanence  of  the  idea  which  he  has  conceived.  By  a  motion  of 
his  hands  which  is  at  once  straight  and  oblique,  consisting  simultaneously 
of  a  pressure  and  a  traction,  he  impresses  on  it  a  motion  which  is,  for 
a  top,  supernatural.  Naturally  the  plaything  would  have  no  other 
motion  than  the  downward  motion  common  to  all  heavy  bodies — the 
child  gives  it  the  opportunity  to  move  circularly,  like  the  Heavens. 
This  motive  spirit,  imparted  by  the  child,  is  invisibly  present  in  the  ma 
terial  of  the  toy — the  length  of  time  for  which  it  remains  there  depends 
on  the  impressive  force  which  communicates  this  property.  When  this 
spirit  ceases  to  animate  the  toy,  it  resumes  its  motion  towards  the  centre, 
as  at  the  beginning.  Do  we  not  have  here  an  image  of  what  happened 
when  the  Creator  wanted  to  give  the  spirit  of  life  to  an  inanimate  body?  " 

1  Dialogus  trilocutorius  de  Possest,  translated  into  French  by  DUHEM. 


72  THE    ORIGINS 

The  World-system  accepted  at  that  time  assumed  that  the  motion 
of  the  different  celestial  spheres  was  maintained  by  that  of  the  outer 
most  sphere,  itself  activated  by  a  Prime  Mover.  Nicholas  of  Cues 
held  that  is  was  sufficient  that  the  Creator  should  have  imparted  an 
impetus  to  the  spheres  at  the  beginning,  and  that  the  impetus  would 
then  be  conserved  indefinitely.  Thus  we  come  across  the  "  chique- 
naude,  "  the  fillip,  of  which  Pascal  talked  in  connection  with  Descartes. 
This  was  also  the  doctrine  of  the  Parisian  Schoolmen  of  the  XlVth 
Century,  of  Buridan  and  Albert  of  Saxony — in  these  celestial  bodies 
there  is  no  influence  which  can  corrupt  the  initial  impetus.  In  a  manner 
which,  for  Nicholas  of  Cues,  is  very  precise,  the  rotational  motion 
of  any  perfect  sphere  is  a  natural  motion.  The  impression  of  impetus 
on  a  moving  body  is  comparable  with  the  creation  of  a  soul  in  the  body. 

Nicholas  of  Cues  became  one  of  the  inspirations  of  Copernicus 
and  of  Kepler,  as  well  as  of  Leonardo  da  Vinci. 

4.  LEONARDO  DA  VINCI'S  CONTRIBUTION  TO  MECHANICS. 

In  mechanics,  Leonardo  da  Vinci  cuts  the  figure  of  a  gifted  amateur. 
Though  he  had  read  and  meditated  upon  the  Schoolmen  that  preceded 
him,  his  bold  imagination  was  not  inhibited  as  theirs  was.  He  tackled 
all  kinds  of  problem,  often  with  more  faith  than  success.  Frequently 
he  returned  to  the  same  problem  by  very  different  paths,  and  did  not 
scruple  to  contradict  himself. 

Leonardo  made  no  concessions  to  systematics.  But  it  seems  that 
the  original  ideas  which  he  threw  off  throughout  his  manuscripts  were 
taken  over  by  more  than  one  of  his  successors. 

His    work    in    mechanics    is    quite 

^        sn n      unique,     and    the    few    pages     which 

we  are  able  to  devote,  in  this  book,  to 
an  attempt  at  analysing   its   objective 
content  can  only  provide  a  feeble  echo 
of  the  torrent  of  ideas  which  flowed  from 
this    "  autodidacte l    par     excellence." 
a)  Leonardo    da   Vinci's   concept    of 
lg*  19  moment,  —  Leonardo  da  Vinci  grasped 

the  idea  of  moment  and  applied  it  in 
a  most  complete  way  to  a  heavy  body  turning  about  a  horizontal  axis, 
a  body  which  he  described  as  being  "  convolutable.  "  Thus,  for  a  lever 
nb  turning  about  a  point  n,  Leonardo  stated  the  following  rule. 

1  Autodidacte  =  one  who  is  self-taught. 


XVth   AND   XVIth   CENTURIES 


73 


"  The  ratio  of  the  distance  (length)  mn  to  the  distance  nb  is  such 
that  it  is  also  the  ratio  of  the  falling  weight  at  d  to  the  (same)  weight 
at  the  position  6.  " l 

That  is,  the  effect  of  a  heavy  body  suspended  at  d  is  the  same  as 
if  the  body  were  suspended  from  the  arm  of  the  horizontal  lever,  nm, 
that  is  obtained  by  projecting  nd  onto  the  horizontal  nb.  Leonardo 
called  this  arm  of  a  horizontal  lever,  equivalent  to  the  inclined  arm  rad, 
the  "  arm  of  the  potential  lever.  "  It  would  seem  here  that  he  had  read 
the  XHIth  Century  statists.2 

b)  The  motion  of  a  heavy  body  on  an  inclined  plane.  —  This  is  one 
of  the  problems  that  captured  Leonardo  da  Vinci's  interest,  and  which 
evoked  some  rather  strange  arguments  from  him.  Thus  we  find  the 
following  passage  among  his  writings. 


Fig.  20 

"  A  heavy  spherical  body  will  assume  a  motion  which  is  all  the 
more  rapid  as  its  contact  with  its  resting  place  is  further  separated 
from  the  perpendicular  through  its  central  line.  The  more  ab  is  shorter 
than  ac,  the  more  slowly  the  ball  will  fall  along  the  line  ac  [than  along 
the  vertical  at]  . .  .  because,  if  p  is  the  pole  of  the  ball,  the  part  m 
which  is  outside  p  would  fall  more  rapidly  if  there  were  not  that  small 
resistance  provided  by  the  counterpoising  of  the  part  o.  And  if  there 
were  not  this  counterpoise,  the  ball  would  fall  along  the  line  ac  more 
quickly  if  o  divided  into  m  more  often.  That  is,  if  the  part  o  divides 
into  m  one  hundred  times,  and  the  part  o  is  missing  throughout  the 

1  Les  Manuscrits  de  Leonard  de   Find,  published  by  Ch. 
Paris,  1890,  Ms.  E,  fol.  72. 

2  See  above  p.  42. 


74 


THE    ORIGINS 


rotation  of  the  ball,  this  will  fall  more  quickly  on  n  by  one  hundredth 

of  the  ordinary  time Iff  is  the  pole  at  which  the  ball  touches  the  plane, 

the  greater  the  distance  between  n  and  p,  the  more  rapid  the  ball's  journey 

will  be.  "  i 

Elsewhere  Leonardo  wrote  on  the  same  subject  in  the  following  terms. 

"  On  motion  and  weight.  All  heavy  bodies  seek  to  faU  to  the  centre, 
and  the  most  oblique  opposition  provides  the  smallest  resistance. 

"  If  the  weight  is  at  A9  its  true  and  direct  resistance  will  be  AB. 
But  the  pole  is  at  the  place  where  the  circumference  touches  the  earth, 
and  the  portion  which  is  furthest  outside  the  pole  falls.  If  SX  is  the 
pole,  it  is  clear  that  ST  will  weigh  more  than  SR,  from  which  it  follows 
that  the  part  ST  falls,  that  it  dominates  over  SI?  and  lifts  it  up,  and 
then  moves  along  the  slope  with  fury.  If  the  pole  were  at  IV,  the 
more  often  AN  divided  into  AC,  the  more  quickly  the  wheel  would 
run  along  the  slope  than  if  it  were  at  X.  "  2 


Fig.  21 

I  am  reluctant  to  comment  on  these  texts  and  to  attribute  to  Leo 
nardo  things  that  he  did  not  intend.  Certainly  he,  like  the  aristotelianB, 
did  not  differentiate  between  dynamics  and  statics.  It  is  also  true 
to  say  that  he  reproduced  and  repeated  the  law  of  powers  which  Aris 
totle  had  formulated  (see  above,  page  20).  Further,  Duhem,  arguing 
from  the  relation  of  velocities  that  Leonardo  gave,  believes  that  he  im 
mediately  applied  this  same  relation  to  powers — that  is,  to  the  apparent 
weights  of  a  given  body  on  differently  inclined  planes — and  that  he 
arrived  in  this  way  at  the  accurate  law  which  we  now  accept. 

1  Ms.  A,  fol.  52. 

2  Ibid.,  fol.  21. 


XVth   AND   XVIth   CENTURIES 


75 


I  believe  that  it  is  more  accurate  to  take  a  view  of  this  kind — that 
Leonardo  only  sketched  the  solution  of  a  problem  which  his  rich  im 
agination  had  formulated,  but  that  he  never  gave  it  a  final  form. 
It  certainly  seems  that  Leonardo  was  unaware  of  the  solution  of  the 
same  problem  that  the  unknown  author  of  Liber  Jordani  de  ratione 
ponderis  had  given,  and  which  was  based  on  the  single  concept  of 
gravitas  secundum  situm,  considered  as  a  first  principle  of  the  statics 
of  heavy  bodies.1 


' m 


Fig.  22 


Leonardo  has,  moreover,  the  merit  of  having  attempted  to  solve 
this  same  problem  of  the  inclined  plane  by  another  method,  one  which 
was  unknown  to  his  predecessors.  Thus  he  observes2  that  a  uniform 
heavy  body  which  falls  obliquely  divides  its  weight  into  two  different 
aspects,  along  the  line  be  and  along  the  line  nm.  But  here  again  we 
are  left  in  suspense — he  does  not  carry  out  the  resolution  of  the  weight 
into  its  two  components  along  be  and  nm  (normal  to  be). 

c)  Leonardo  da  Vinci  and  the  reso 
lution  of  forces.  —  Leonardo  asked  him 
self  how  the  weight  of  a  heavy  body,  sup 
ported  by  two  strings,  was  apportioned 
between  these  two.  He  was  of  the 
opinion  that  the  weight  of  the  body 
suspended  at  b  was  divided  between  the 
strings  bd  and  ba  as  the  ratio  of  the  lengths 
ea  and  de.  This  guess  contradicts  the  now 
classical  rule  of  the  parallelogram. 

However  Leonardo  used,  at   least  implicitly,  the  following  rule. 

"  With  respect  to  a  point  taken  on  one  of  the  components  of  a 
force,  the  moment  of  the  other  component  is  equal  to  the  moment  of 
the  total  force  with  respect  to  the  same  point.  " 


Fig.  23 


1  See  above,  p.  43. 
*  Ms.  G,  fol.  75. 


76 


THE    ORIGINS 


Fig.  24 


Thus,  through  the  intermediary  of  the  concept  of  moment,  Leonardo 
arrived  at  the  resolution  of  forces.  Indeed,  on  different  occasions  he 
drew  the  figure  opposite,  in  which  the  weight  JV  is  hung  from  two 

strings  CB,   CA,  which  are  equally   inclined 
to  the  vertical  through  JV.     He  wrote — 

"  The  pole  of  the  angular  balance  formed 
of  AD  and  AF  is  A9  and  its  appendages  are 
DN  and  FC. 

"  As  the  angle  of  the  string  that  carries 
the  weight  N  at  its  centre  increases,  the 
length  of  its  potential  lever  decreases  and 
the  length  of  the  potential  counter-lever 
which  carries  the  weight  increases.  " 

This  remains  somewhat  mysterious,  but, 
like  Duhem,1  one  may  believe  that  the  ten 
sion  of  the  string  CB,  and  the  weight  IV, 
would  maintain  the  rigid  body  formed  of  the 
two  potential  arms  AB  and  AF  in  equili 
brium,  if  the  body  were  able  to  turn  about  the  point  A. 

"  A  confusion  of  ideas  poured  from  Leonardo's  mind  but,  to  a  high 
degree,  he  lacked  the  power  of  discriminating  between  the  true  and 
the  false.  Also,  as  an  inevitable  consequence,  a  truth  which  might 
emerge  from  the  surface  of  incomplete  or  false  beliefs  and  become 
clear  to  him  at  one  instant,  was  thrown  back  again,  to  await  the  future 
which  would  finally  return  it  to  the  shore.  " 

d)  Leonardo  and  the  Energy  of  moving  bodies.  —  Leonardo  was 
aware  of  Buridan's  doctrine  through  the  intermediary  of  Albert  of 
Saxony,  who  had  adopted  it.  Moreover,  he  had  read  Nicholas  of  Cues. 

Leonardo  reconciled  these   doctrines  in  the  following  way. 

At  the  outset  he  defined  a  quantity  called  impeto,  analogous  to 
impetus  in  Buridan's  sense —  "  Impeto  is  a  virtue  created  by  motion 
and  transmitted  from  the  motor  to  the  moving  body,  which  has  as  much 
motion  as  the  impeto  has  life.  "  2 

Elsewhere  he  says,  "  Impeto  is  the  impression  of  motion  which  is 
transmitted  from  the  motor  to  the  moving  body.  .  .  .  All  impression 
desires  permanance,  as  is  shown  us  by  the  similarity  of  the  motion 
impressed  on  the  body.  "  3 

Leonardo  regarded  the  motion  of  a  projectile  as  being  separated 

1  0.  S.,  Vol.  I,  p.  181. 

2  Ms.  E,  fol.  22. 

3  Ms.  G,  fol.  73. 


XVth   AND    XVIth   CENTURIES  77 

into  three  phases.  In  the  first  the  motion  is  purely  violent  and  is 
effected  as  if  the  projectile  had  no  mass  and  was  subject  only  to  the 
initial  impeto. 

In  the  third  period,  the  impeto  has  completely  disappeared.  The 
moving  body  has  a  purely  natural  motion  under  the  sole  influence  of 
gravity. 

Between  these  two  extreme  phases  Leonardo  assumed  the  existence 
of  an  intermediate  period  in  which  the  motion  was  mixed,  part  violent, 
part  natural.  This  is  the  period  of  compound  impeto. 

The  following  quotation  will  illustrate  this  idea. 

"  A  stone  or  other  heavy  thing,  thrown  with  fury,  changes  the 
direction  of  its  travel  half  way  along  its  path.  And  if  you  are  able 
to  shoot  a  cross-bow  for  200  yards,  place  yourself  at  a  distance  of  100 
yards  from  a  tower,  aim  at  a  point  above  the  tower  and  shoot  the 
arrow.  You  will  see  that  100  yards  from  the  tower  the  arrow  will  be 
driven  in  perpendicularly.  And  if  you  find  it  thus,  it  is  a  sign  that 
the  arrow  has  finished  its  violent  motion  and  has  started  the  natural 
motion,  that  is,  that  being  heavy,  it  falls  freely  towards  the  centre.  " x 

Or  better  still— 

"  On  convolutory  motion.  A  top  which  loses  the  power  which  the 
inequality  of  its  weight  has  about  the  centre  of  its  convolution  because 
of  the  speed  of  this  convolution,  because  of  the  effect  of  the  impeto 
which  dominates  the  body,  is  one  which  will  never  have  that  tendency 
to  fall  lower,  which  the  inequality  of  the  weight  seeks  to  do,  as  long 
as  the  power  of  the  body's  motive  impeto  does  not  become  less  than 
the  power  of  the  inequality. 

"  But  when  the  power  of  the  inequality  surpasses  the  power  of 
the  impeto,  then  it  becomes  the  centre  of  the  motion  of  convolution 
and  the  body,  brought  to  a  recumbent  position,  expends  the  remainder 
of  the  aforesaid  impeto  about  this  centre. 

"  And  when  the  power  of  this  inequality  becomes  equal  to  the  power 
of  the  impeto,  then  the  top  is  inclined  obliquely,  and  the  two  powers 
struggle  with  each  other  in  a  compound  motion,  both  moving  in  a  wide 
circuit,  until  the  centre  of  the  second  kind  of  convolution  is  established. 
In  this  the  impeto  expends  its  power. " 2 

Following  the  example  of  Nicholas  of  Cues,  Leonardo  concerned 
himself  with  the  "  game  of  the  sphere  ."  He  wrote — 

"  On  compound  impeto.  A  compound  motion  is  one  in  which  the 
impeto  of  the  motor  and  the  impeto  of  the  moving  body  participate 

1  Ms.  A,  fol.  4. 

2  MH.  K,  fol.  50. 


78 


THE   ORIGINS 


together,  as  in  the  motion  FBC  which  is  intermediate  between  two 
simple  motions.  One  of  these  is  close  to  the  beginning  of  the  motion 
and  the  other  close  to  the  end.  But  the  first  is  determined  solely  by 
the  motor,  and  the  second  only  by  the  shape  of  the  body. 

"  On  decomposed  impeto.  Decomposed 
impeto  is  associated  with  a  moving  body 
which  has  three  kinds  of  impeto.  Two  of 
these  arise  from  the  motor  and  the  third 
arises  from  the  moving  body.  But  the 
two  that  arise  from  the  motor  are  the 
rectilinear  motion  due  to  the  motor  and 
the  curved  motion  of  the  moving  body, 
and  are  mixed  together.  The  third  is 
the  simple  motion  of  the  moving  body, 
which  only  tends  to  turn  round  with  its 
centre  of  convexity  in  contact  with  the 
plane  on  which  it  turns  and  lies.  "  L 

Here  da  Vinci's  imagination  is  given 
free  rein.  Our  author  becomes  even  more 
lyrical  when  he  defines  the  forza. 

"  As  for  the  forza —  I  say  that  the 
forza  is  a  spiritual  quality,  an  invisible 

power  which,  by  means  of  an  external  and  accidental  violence,  is 
caused  by  the  motion  and  introduced,  fused,  into  the  body  ;  so  that  this 
is  enticed  and  forced  away  from  its  natural  behaviour.  The  forza  gives 
the  body  an  active  life  of  magical  power,  it  constrains  all  created  things 
to  change  shape  and  position,  hurtles  to  its  desired  death  and  changes 
itself  according  to  circumstances.  Slowness  makes  it  powerful  and 
speed,  weak — it  is  born  of  violence  and  dies  in  freedom.  The  stronger 
it  is,  the  more  quickly  it  consumes  itself.  It  furiously  drives  away 
anything  that  opposes  it  until  it  is  itself  destroyed— it  seeks  to  defeat 
and  kill  anything  that  opposes  it  and,  once  victorious,  dies.  It  be 
comes  more  powerful  when  it  meets  great  obstacles.  Every  thing 
willingly  avoids  its  death.  All  things  which  are  constrained  constrain 
themselves.  Nothing  moves  without  it.  A  body  in  which  it  is  born 
does  not  increase  in  weight  or  size.  No  motion  that  it  creates  is  lasting. 
It  grows  in  exertion  and  vanishes  in  rest.  A  body  on  which  it  is  im 
pressed  is  no  longer  free.  "  2 

Or  again,  "  I  say  that  forza  is  a  spiritual,  incorporeal,  invisible  power 
which  is  created  in  bodies  which,  because  of  an  accidental  violence,  are 

1  Ms.  E,  fol.  35. 

2  Ms.  A,  fol.  35. 


Fig.  25 


XVth   AND   XVIth   CENTURIES  79 

in  some  other  state  that  their  natural  being  and  rest,  I  have  said 
spiritual  because  there  is  in  this  forza  an  active  incorporeal  life,  and  I 
have  said  invisible  because  bodies  in  which  it  is  born  change  neither  in 
weight  nor  in  shape  ;  of  short  life,  because  it  always  seeks  to  overcome 
its  cause,  and  having  done  so,  to  die.  "  1 

We  shall  attempt,  without  too  much  quotation,  to  indicate  Leo 
nardo's  ideas  on  forza,  ideas  which  were  inspired  by  the  metaphysics  of 
Nicholas  of  Cues. 

Forza  can  be  born  of  the  "  expansion  undergone  by  a  tenuous  body 
in  one  that  is  dense,  like  the  multiplication  of  fire  during  the  firing  of 
cannons.  "  It  can  also  be  born  of  a  deformation  as  in  a  cross-bow. 
Finally,  one  forza  can  engender  another — this  is  the  case  of  impact. 

Leonardo  returned  to  a  pythagorean  doctrine  according  to  which  a 
heavy  body  that  is  detached  from  a  star  to  which  it  belongs  tends  to 
return  there,  in  order  to  reconstitute  the  completeness  of  the  star.  He 
contrasted  weight  with  forza,  saying  that  these  oppose  each  other. 

"  Weight  is  natural  and  seeks  stability,  then  rest — forza  seeks 
killing  and  death  for  itself.  "  Weight  is  indestructible.  When  a  heavy 
body  arrives  on  the  ground  it  exerts  a  pressure  on  it,  "  and  penetrates, 
from  one  support  to  another,  to  the  centre  of  the  World.  "  A  weight 
embodies  power,  /orza,  motion  and  impact  at  the  same  time.  But  the 
fall  of  a  body  is  itself  preceeded  by  an  accidental  ascent.  To  be  precise, 
at  the  origins  of  all  actions  in  mechanics  there  must  be  a  prime  mover. 
And  Leonardo,  seduced  by  metaphysics,  concludes — all  motion  arises 
from  the  mind. 

Further  comment  on  this  adventurous  thesis  of  Leonardo  seems,  to 
us,  unnecessary — its  qualities  are  more  of  poetry  than  of  precision,  of 
eloquence  than  solidity,  more  metaphysical  than  positive. 

e)  Leonardo  da  Vinci  and  perpetual  motion.  —  Leonardo  denied  the 
possibility  of  perpetual  motion  on,  the  grounds  that  forza  continually 
expends  itself.     On  the  other  hand,  gravity  seeks  to  produce  equili 
brium,  all  motions  which  are  set  in  train  by  gravity  have  rest  as  their 
ultimate  end. 

f )  Leonardo  and  the  Figure  of  the  Earth.  —  Having  read  and  medi 
tated  Albert  of  Saxony,  Leonardo  wrote  in  connection  with  the  figure  of 
the  Earth — 

"  Every  heavy  body  tends  downwards,  and  things  which  are  at  a 
height  will  not  remain  there,  but  will  all,  in  time,  fall  down.  Thus,  in 
time,  the  World  will  become  spherical  and  in  consequence,  will  be  com- 

1  Ms.  B,  fol.  63. 


80 


THE    ORIGINS 


pletely  covered  with  water.  " l  And,  without  hesitation,  he  adds, — the 
Earth  will  be  uninhabitable. 

In  Leonardo's  belief,  the  seas  exerted  no  pressure  on  the  part  of  the 
globe  which  they  covered.  Quite  the  contrary.  "  A  heavy  body 
weighs  more  in  a  lighter  medium.  Therefore  the  Earth,  that  is  covered 
by  air  is  heavier  than  that  covered  by  water.  "  2 

g)  Leonardo  da  Vinci  and  the  theory 
of  centre  of  gravity.  The  flight  of  birds.  — 
Leonardo  considered  two  towers  ABIQ, 
CDL/C  in  "  continual  uprightness, " 
erected  parallel  to  each  other  from  the 
bases  AB,  CD,  on  the  Earth.  He  pre 
dicted  that  "  the  two  towers  will 
tumble  down  towards  each  other  if  their 
construction  is  continued  above  a  cer 
tain  height  in  each  case.  " 

Here  is  his  argument.  "  Let  the 
two  verticals  through  B  and  C  be  pro 
duced  in  c  continual  straightness.  '  If 
they  cut  one  of  the  towers  in  GC  and  the 
other  in  jBF,  it  follows  that  these  lines 
do  not  pass  through  the  centre  of  gravity 
of  the  lengths  of  the  towers.  Therefore 

KLCG,  a  part  of  one  tower,  weighs  more  than  the  remainder,  GCD, 
and  of  these  unequal  things,  one  will  be  dominant  over  the  other,  in 
such  a  way  that,  of  necessity,  the  greatest  weight  of  the  tower  will  carry 
away  all  the  opposite  tower.  And  the  other  tower  will  do  the  same,  in  a 
way  which  is  inverse  to  the  first. "  3 

To  recapitulate,  Leonardo  asserted  that  the  vertical  from  the  centre 
of  gravity  should  not  pass  outside  the  base.  This  is,  implicitly,  the  now 
classical  theorem  of  the  polygon  of  sustentation,  but  it  contains  the 
error,  common  to  all  the  Schoolmen,  that  the  convergence  of  the  ver 
ticals  has  not  been  neglected. 

In  this  connection,  Leonardo  almost  goes  as  far  as  to  suggest  that  a 
measurement  of  the  distance  apart  of  two  verticals  at  the  top  and  the 
bottom  of  a  tower  should  be  used  to  deduce  the  length  of  the  Earth's 
radius.4 


Fig.  26 


1  Ms.  F,  foL  70. 

2  Ibid.,  fol.  69. 

3  Ibid.,  fol.  83. 

4  DUHEM,  0.  S.,  Vol.  II,  p.  81, 


XVth   AND    XVIth    CENTURIES  81 

Going  over  from  statics  to  dynamics,  Leonardo,  guided  by  his  bold 
and  ubiquitous  imagination,  affirmed  that  "  any  heavy  body  moves 
towards  the  side  on  which  it  weighs  more.  ,  .  .  The  heaviest  parts  of 
bodies  which  move  in  air  become  guides  for  their  motion.  "  x 

He  also  wrote,  "  Every  thing  which  moves  on  a  perfectly  plane 
ground  in  such  a  way  that  its  pole  is  never  found  between  parts  of  equal 
weight,  never  comes  to  rest.  An  example  is  provided  by  those  who 
slide  on  ice,  and  who  never  stop  if  their  parts  do  not  become  equidistant 
from  their  centres.  "  2 

In  his  Treatise  on  Painting,  Leonardo  applied  the  preceding  ideas 
to  the  flight  of  birds. 

66  Any  body  that  moves  by  itself  will  do  so  with  greater  velocity  if 
its  centre  of  heaviness  is  further  removed  from  its  centre  of  support. 

"  This  is  mentioned  principally  in  connection  with  the  motion  of 
birds.  These,  without  any  clapping  of  wings  or  assistance  from  the 
wind,  move  themselves.  And  this  occurs  when  the  centres  of  their 
heaviness  are  displaced  from  the  centres  of  their  support,  that  is,  away 
from  the  middle  of  the  extension  of  their  wings.  Because,  if  the  middle 
of  the  two  wings  is  in  front  of  or  behind  the  middle,  or  the  centre,  of 
the  heaviness  of  the  whole  bird,  then  the  bird  will  carry  its  motion 
upwards  or  downwards  [and  this  all  the  more  so]  as  the  centre  of  heavi 
ness  is  more  distant  from  the  middle  of  the  wings.  ..." 

h)  Leonardo  and  the  fall  of  bodies.  —  It  was  inevitable  that  Leonardo 
should  have  become  interested  in  the  fall  of  heavy  bodies.  After  having 
hesitated  for  some  time  between  the  two  laws  of  velocity  that  were 
mentioned  by  Albert  of  Saxony  (see  above,  page  57),  Leonardo  declared 
himself  entirely  in  favour  of  the  correct  law  v  =  kt .  To  set  against  this, 
the  content  of  the  studies  on  the  latitude  of  forms  (Oresme,  Heytesbury) 
completely  escaped  him.  Throughout  he  believed  that  motion  (moto) 
was  proportional  to  velocity  (vclocitas)  and,  in  consequence,  was 
mistaken  about  the  law  of  distances. 

In  this  connection  we  shall  confine  ourselves  to  a  single  quotation. 

"  On  motion.  A  heavy  body  which  falls  freely  acquires  one  unit  of 
motion  in  each  unit  of  time  ;  and  one  unit  of  velocity  for  each  unit  of 
motion. 

"  Let  us  say  that  in  the  first  unit  of  time  it  acquires  one  unit  of 
velocity.  In  the  second  unit  of  time  it  will  acquire  two  units  of  motion 
and  two  units  of  velocity,  and  so  on  in  the  way  described  above.  "  3 

1  Ms.  Et  fol.  57. 

2  Ms.  A,  fol.  21. 

3  Ms.  M,  fol.  45. 


82  THE    ORIGINS 

i)  Leonardo's  hydrostatics.  —  Like  the  ancients,  Leonardo  set  out  to 
explain  now  water  could  appear  in  springs  at  the  tops  of  mountains. 
He  wrote,  "  It  must  be  that  the  cause  which  keeps  blood  at  the  top  of 
a  man's  head  is  the  same  as  that  which  keeps  water  at  the  tops  of 
mountains.  "  Leonardo  sought  this  mechanism  in  the  nature  of  heat, 
"  There  are  veins  which  thread  throughout  the  body  of  the  Earth.  The 
heat  of  the  Earth,  distributed  throughout  this  continuous  body,  keeps 
the  water  raised  in  these  veins  even  at  the  highest  summits.  " l  To  be 
accurate,  Albert  of  Saxony,  in  his  commentary  of  the  relevant  parts  of 
Aristotle's  treatise  Meteores,  had  already  invoked  the  intervention  of 
heat  in  this  matter. 

Leonardo  was  more  fortunate  when  he  gave  a  complete  formulation 
of  the  law  of  the  flow  of  currents. 

"  All  motion  of  water  of  uniform  breadth  and  surface  is  stronger  at 
one  place  than  at  another  according  as  the  water  is  shallower  there  than 
at  the  other.  "  Leonardo  also  outlined  a  theory  of  hydraulic  pumps  in 
the  writing  Del  moto  e  misura  delVacqua  in  which  a  hint  of  Pascal's 
principle  can  be  discovered.2 

j)  Leonardo  da  Vinci  and  the  geocentric  hypothesis.  —  On  looking  for 
it  in  Leonardo's  writings,  there  can  always  be  found  evidence  of  the 
kind  that  Duhem  indefatigably  sought.  Thus  there  is  the  following 
passage,  which  is  aimed  at  the  geocentric  hypothesis.  "  ,  ,  .  Why  the 
Earth  is  not  at  the  centre  of  the  circle  of  the  Sun  nor  at  the  centre  of  the 
World,  but  rather  at  the  centre  of  its  elements,  which  accompany  it 
and  with  which  it  is  united.  "  3 


5.  NICHOLAS  COPERNICUS  (1472-1543).    His  SYSTEM  OF  THE  WORLD 

AND  HIS  IDEAS  ON  ATTRACTION. 

In  this  book  we  can  only  discuss  the  different  World-systems  to  the 
extent  that  they  have  had  an  influence  on  the  development  of  mechanics. 
The  copernican  system  that  was,  in  the  hands  of  Kepler  and  Newton, 
to  play  a  fundamental  part  in  the  creation  of  dynamics,  had  no  imme 
diate  influence  on  the  scientists  of  the  Renaissance.  On  the  whole, 
these  remained  faithful  to  aristotelian  ideas.  We  remark,  for  example, 
that  the  Sorbonne  in  the  XVIth  Century  remained  closed  to  copernican 
ideas  and  continued  to  teach  Ptolemy's  system. 

Our  attention,  therefore,  should  only  be  held  for  a  short  time  by 

1  Ms.  A,  fol.  56. 

2  Cf.  DUHEM,  Etudes  sur  Leonard  de  VincL  Series  I,  t>    198 

3  Ms.  F,  fol.  41.  * 


XVth  AND   XVIth    CENTURIES  83 

Copernicus'  ideas  on  dynamics  and  the  circumstances  which  facilitated 
the  copernican  revolution. 

From  Antiquity  there  had  been  writers  whose  opinions  were  similar 
to  those  of  Copernicus.  Philolaus  of  Crete  (a  disciple  of  Pythagoras), 
Nicete  of  Syracuse  and  Aristarchus  of  Samos  had  attributed  to  the 
Earth  both  a  daily  and  an  annual  motion,  circular  and  oblique,  about 
the  Sun.  (There  was  also  supposed  to  an  invisible  earth  which  was 
symmetrical  with  ours  with  respect  to  the  Sun.) 

In  the  Middle  Ages  William  of  Ockham,  Buridan  and  Albert  of 
Saxony  assumed  that  the  Earth  could  have  a  rotational  motion  which 
was  not  necessarily  identical  with  the  apparent  motion  of  the  stars. 
Albert  of  Saxony  was  not  alone  in  attributing  the  precession  of  the 
equinoxes  to  a  slow  displacement  of  the  Earth.  We  have  seen  in  detail 
how  Nicole  Oresme,  who  was  certainly  unknown  to  Copernicus,  had 
defended  the  theories  of  a  fixed  Heaven  and  an  Earth  which  had  a 
diurnal  motion.  The  appeal  to  the  doctrine  of  impetus  in  Oresme's  thesis, 
which  was  used  to  destroy  that  of  Aristotle,  is  especially  important. 

In  the  religious  fiels  the  Church  in  the  XHIth  Century,  tolerant 
because  of  its  power,  had  the  wisdom  to  brush  aside  the  a  priori  questions 
which  could  be  opposed  to  every  doctrine  that  deviated  from  the  geo 
centric  hypothesis.  As  early  as  1277  Etienne  Tempier,  Bishop  of  Paris, 
made  the  assumption  that  the  question  of  whether  the  Heavens  had  a 
translation  motion,  or  not,  could  be  discussed.  Thus  the  Church  in  the 
XHIth  Century  assumed  that  the  study  of  world-systems  could  be 
pursued  as  a  piece  of  contingent  research.  In  fact,  a  century  later, 
Nicole  Oresme  did  not  compromise  his  ecclesiastical  career  in  any  way 
by  believing  in  the  motion  of  the  Earth. 

In  the  field  of  metaphysics,  even  the  debate  on  the  plurality  of 
worlds  helped  the  copernican  revolution.  In  reaction  against  the  pytha- 
gorean  doctrine,  Aristotle  explicitly  understood  the  term  "  Heavens  " 
(OvQKv6$)  in  the  sense  of  "  All  "  or  of  "  Universe. "  The  absolute 
fixity  of  the  Earth  and  the  perpetual  rotation  of  the  Heavens  consti 
tuted  a  dogma  of  science.  The  Universe  is  unique  and  each  body  has 
a  unique  natural  place,  to  which  it  returns  of  its  own  accord  if  it  is 
violently  displaced  from  it.  Any  other  world  which  can  exist  must 
necessarily  be  made  of  the  same  elements  as  ours*  To  Aristotle,  this 
meant  that  the  co-existence  of  several  worlds  implied  a  contradiction. 
Beyond  this  Eight  Sphere  there  can  be  neither  space  nor  time. 

This  thesis  was  to  be  attacked  in  the  Xlllth  Century  on  the  very 
grounds  of  the  omnipotence  of  God. 

Michael  Scot  (1230)  was  of  the  opinion  that  God  could  have  created 
several  worlds,  but  that  Nature  would  not  have  been  able  to  accommodate 


84  THE    ORIGINS 

them.  Saint  Thomas  Aquinas  attempted  to  reconcile  Aristotle's  doc 
trine  with  the  principle  of  divine  omnipotence — the  creation  of  similar 
worlds  would  be  superfluous,  the  creation  of  dissimilar  worlds  would 
detract  from  the  perfection  of  each  of  them,  for  only  the  ensemble  can 
be  perfect. 

In  1277  the  theologians  of  Paris,  at  the  request  of  Etienne  Tempier, 
condemned  the  anti-pluralist  thesis. 

William  of  Ockham  intervened  in  the  same  direction — he  argued  that 
identical  elements  could  simultaneously  be  directed  towards  different 
places.  Thus  a  fire  at  Oxford  would  not  move  towards  the  same  place 
as  if  it  had  been  lit  at  Paris.  The  direction  of  a  natural  motion  could 
therefore  depend  on  the  initial  position  of  the  element.  Albert  of 
Saxony  decided  against  the  plurality  of  worlds  except  u  in  a  super 
natural  way,  to  the  liking  of  God.  "  On  the  other  hand,  towards  the 
end  of  the  XVth  Century  Joannes  Majoris  asserted,  in  his  De  infinite, 
not  only  the  plurality  of  worlds  but  the  existence  of  an  infinite  number 
of  worlds. 

These  discussions  in  no  way  lessen  the  originality  of  Copernicus' 
work,  but  to  a  certain  extent  they  explain  why  he  ventured  to  present 
his  thesis.  Being  by  profession  Canon  of  Thorn,  he  protected  himself 
with  certain  cautious  declarations.  Thus,  in  dedicating  his  works  to 
Pope  Paul  III,  he  wrote,  "  I  have  believed  that  I  would  be  readily 
permitted  to  examine  whether,  in  supposing  the  motion  of  the  Earth, 
something  more  conclusive  (firmiores  demonstrations)  might  not  be 
found  in  the  motion  of  the  celestial  bodies.  " 

Strictly  speaking,  the  doctrinal  opposition  only  came  much  later 
with,  for  example,  Melanchton  and  Father  Riccioli.  The  latter  was 
able  to  enumerate  77  arguments  against  the  motion  of  the  Earth  and  to 
refute  49  of  the  copernican  arguments.  As  far  as  the  Congregation  of 
Cardinal  Inquisitors  was  concerned,  it  only  officially  condemned  Coper 
nicus'  writings  on  March  5th,  1616.  In  order  to  fix  certain  essential 
dates,  we  recall  that  Copernicus  was  born  at  Thorn  on  January  19th, 
1472.  He  received  his  doctorate  at  Krakov,  and  made  his  way  to 
Bologna  and  then  to  Rome,  where  he  devoted  himself  to  astronomy. 

Copernicus  gave  himself  up  to  a  thorough  study  of  the  different 
world-systems  which  had  been  proposed  by  the  Ancients,  and  used  the 
motions  of  Mercury  and  Venus  in  order  to  place  the  Sun  at  the  centre  of 
the  planets.  In  referring  to  the  Pythagoreans,  he  proposed  that  the 
Sun  should  be  placed  at  the  centre  of  the  World.  Not  wishing  to  ad 
vance  anything  without  evidence,  he  started  observation  of  planetary 
motions.  The  account  of  this  task,  completed  in  1530,  was  only  printed 
on  his  death  in  1543, 


XVth   AND    XVIth    CENTURIES  85 

If,  in  not  making  the  centre  of  the  Earth  coincide  with  that  of  the 
Universe,  he  dispensed  with  the  aristotelian  doctrine  on  an  essential 
point,  he  kept,  for  the  rest,  most  of  the  ideas  of  the  Schoolmen.  How 
ever,  he  did  dispose  of  the  distinction  which  Albert  of  Saxony  had  made 
between  the  centre  of  gravity  and  the  geometrical  centre  of  the  Earth. 
We  shall  quote  from  Copernicus'  De  revolutionibus  orbium  caelestium. 

"  The  Earth  is  spherical  because,  on  all  sides,  it  strives  towards  the 
centre.  The  element  of  the  Earth  is  the  heaviest  of  all,  and  all  heavy 
bodies  are  carried  towards  it  and  seek  its  intimate  centre. 

"  To  my  mind,  gravity  is  nothing  else  than  a  certain  natural  quality 
given  to  the  parts  of  the  Earth  by  the  divine  providence  of  He  who  made 
the  Universe,  in  order  that  they  should  converge  towards  their  unity 
and  integrity,  by  uniting  in  the  form  of  a  globe.  It  is  probable  that 
this  property  also  belongs  to  the  Sun,  the  Moon  and  to  the  wandering 
lights  so  that  these  too,  by  its  virtue,  keep  that  round  shape  in  which 
we  see  them.  " 

And  here  Copernicus  attacks  Albert  of  Saxony. 

"  Because  of  their  gravity,  water  and  earth  both  tend  towards  the 
same  centre. . . .  One  should  not  heed  the  Aristotelians  when  they  claim 
that  the  centre  of  gravity  is  separate  from  the  geometrical  centre. . . . 
It  is  clear  that  both  earth  and  water  strive  towards  a  unique  centre  of 
gravity  at  the  same  time,  and  that  this  centre  is  in  no  way  different  from 
the  centre  of  the  Earth.  " 

Copernicus'  doctrine  on  the  figure  of  the  Earth  agreed  perfectly 
with  all  the  geographical  observations.  More  simple  than  that  of 
Albert  of  Saxony — an  abstraction  founded  on  prejudices  opposed  to  the 
motion  of  the  Earth — it  was  destined  to  triumph.  But  at  this  point 
the  copernican  ideas  came  up  against  a  scholastic  tradition  whose  root 
must  be  found  in  Aristotle's  Meteores — the  four  elements  earth,  water, 
air  and  fire  have  equal  masses  and  therefore  occupy  volumes  which  are 
inversely  proportional  to  their  densities.  Moreover,  the  Aristotelians 
held  that  when  a  given  mass  of  an  element  became  "  corrupted  **  in 
order  to  produce  the  next  element  in  the  succession,  its  volume  increased. 
Aristotle  himself  mentioned  this  relationship  for  the  single  instance 
of  the  transformation  of  water  into  air,  but  his  commentators  applied 
it  without  hesitation  to  the  transformation  of  earth  into  water  and, 
carrying  the  argument  to  the  limit,  said  that  the  total  volume  of  water 
was  greater  than  the  total  volume  of  earth.  Gregory  Reisch,  prior  of 
Fribourg,  put  forward  opinions  of  this  kind  in  his  Margarita  philosophica 
(1496),  a  small  encyclopedia  that  was  widely  circulated  in  the  XVIth, 
Century.  Twelve  years  after  the  end  of  Magellan's  navigation  of  the 
globe — which  should  have  clarified  the  scholastic  opinion  of  the  face 


86  THE    ORIGINS 

of  the  Earth — Mauro  of  Florence  (1493-1566)  took  up  Reisch's  thesis 
again,  and  held  that  the  volume  of  the  closed  earth  was  ten  times  less 
than  that  of  the  waters.  Copernicus  felt  himself  obliged  to  refute  this 
author.  We  have  only  stressed  this  geophysical  issue  in  order  to  show 
the  kind  of  objection  which  the  great  reformer  of  the  system  of  the  world 
met  during  his  lifetime. 


6.  JOHN  FERNEL  (1497-1558)  AND  THE  FIGURE  OF  THE  EARTH. 

John  Fernel,  chief  physician  to  Henry  II,  deserves  to  be  mentioned 
in  a  history  of  mechanics  for  having  been  the  first  among  the  moderns 
who  had  the  initiative  to  measure  a  degree  of  terrestrial  meridian.  This 
he  did  by  counting  the  number  of  revolutions  of  the  wheels  of  his  car 
riage  between  Paris  and  Amiens.  In  his  Cosmotheoria,  published  at 
Paris  in  1528,  Jean  Fernel  disputed  Albert  of  Saxony's  doctrine,  and 
decided  in  favour  of  the  existence  of  a  unique  spherical  surface  for  the 
combined  mass  of  earth  and  water.  He  imagined  the  Earth  to  be  abso 
lutely  immobile  and  to  present  the  shape  of  a  globe  which  had  been 
hollowed  out  in  places  and  whose  cavities  had  been  filled  with  water. 

If  one  is  to  believe  the  chronicles,1  John  Fernel,  who  was  a  distin 
guished  astronomer  and  mathematician,  would  certainly  have  written 
other  things  "  if  his  wife  had  not  compelled  him,  so  to  speak,  to  leave 
the  sterile  study  of  mathematics.  " 


7.  ITALIAN  SCHOLASTICISM  IN  THE  xvith  CENTURY. 

At  the  beginning  of  the  XVIth  Century  the  Italian  Schoolmen  were 
divided  into  three  camps  ;  there  were  the  Averroists,  the  Alexandrists 
— those  who  made  appeal  to  Alexander  of  Aphrodisias — and  the  Huma 
nists. 

As  an  example  of  the  first  school,  we  shall  cite  Agostino  Nifo,  to 
whom  we  owe  a  commentary  of  De  Caelo  et  Mundo  dated  1514.  In  this 
manuscript  there  is  nothing  but  scorn  for  the  parisian  school  of  the 
XlVth  Century,  whose  representatives  are  called  Juniores,  terminalista 
(nominalists),  Sorticoles  (disciples  of  Sortes,  that  is,  of  Socrates)  and 
Captiunculatores  (a  corruption  of  Calculators).  Albert  of  Saxony  is 
ridiculed  with  the  title  of  Albertutius  or  Albertus  Parvus. 

The  Averrolsts  rejected  the  doctrine  of  impetus  and  returned  to 
Aristotle's  explanation  of  the  motion  of  projectiles.  Concerning  the 

1  LALANDE,  Astronomic,  Vol.  1,  p.  189. 


XVth  AND    XVIth   CENTURIES  87 

fall  of  heavy  bodies,  Nifo,  like  Saint  Thomas  Aquinas,1  held  that  proxi 
mity  to  its  natural  place  contributed  towards  a  body's  acceleration. 
He  added  to  this  an  "  instrumental  cause,  "  and  a  quality  belonging 
to  the  moving  body. 

Among  the  Alexandrists  Peter  Pomponazzi  of  Mantua,  who  is  said 
to  have  taught  at  Bologna,  devoted  a  treatise  De  intensione  et  rernissione 
formarum  (1514)  to  an  attack  on  the  Oxford  School.  In  his  De  reactione 
(1515)  he  called  William  Heytesbury  "  the  greatest  of  the  Sophists  " 
and  contrasted  him  with  "  the  clear  and  great  voice  of  Aristotle.  " 

The  thesis  of  Alexander  of  Aphrodisias,  to  whom  this  faction  gave 
allegiance,  has  been  preserved  for  us  by  Simplicius.  It  consisted  of  the 
assumption  that  a  heavy  body  which  was  placed  at  a  height  became 
lighter.  This  lightness  obtained  at  the  beginning  of  the  fall  and  then, 
continuously,  became  less  apparent. 

The  Italian  Humanists  reproached  the  Schoolmen  for  their  "  parisian 
manner, "  which  was  described  as  barbarous,  sordid,  gross  and  uncul 
tured,  but  approved  of  their  religious  orthodoxy.  In  addition  the 
Humanists,  in  the  person  of  Giorgio  Valla  who  taught  at  Padua  (1470) 
and  at  Venice  (1481),  made  an  especial  attack  on  the  AverroXsts.  These 
were  taken  to  task  for  their  language — studded  with  arabic  terms — and 
for  their  exclusive  cult  of  Aristotle  and  consequent  neglect  of  Plato. 
Valla  went  as  far  as  to  consider  Averroes,  in  Latin,  of  course,  as  a 
"  primitive  creature  emerging  from  the  mud  "  and  as  "  pigheaded.  " 
In  dynamics  Valla  echoed  the  thesis  of  intermediate  rest  (quies  inter" 
media)  between  the  ascent  and  the  descent  of  a  body,  which  compro 
mised  the  continuity  of  the  motion.  He  assumed  the  existence,  in 
every  moving  body,  of  a  vis  insita*  This  quantity,  however,  has  no 
connection  with  Buridan's  impetus,  but  rather  is  accounted  for  by  the 
proximity  of  a  motive  agency  or  the  natural  place,  according  as  a  violent 
or  a  natural  motion  is  in  question. 

These  violent  polemics  added  nothing  new  to  mechanics,  and  we 
have  only  described  them  in  order  to  illustrate  the  atmosphere  of  the 
time,  with  which  original  thinkers  had  to  contend. 

8.  PARISIAN  SCHOLASTICISM  IN  THE  xvith  CENTURY. 

The  teachings  of  Buridan  and  Albert  of  Saxony  were  preserved  at 
the  College  of  Montaigu  under  the  Scotsmen  Joannes  Majoris  and 
George  Lockhart.  Jean  Dullaert  de  Gand  and  the  Spaniard  Luiz 
Coronel  taught  at  the  same  college.  Another  Spaniard,  Jean  de  Celaya, 
taught  at  Sainte-Barbe. 

1  See  above,  p.  57. 


0«  THE    ORIGINS 

This  tradition  was  eclectic.  It  declared  that  it  followed  "  the  triple 
voice  of  Saint  Thomas  Aquinas,  the  Realists  and  the  Nominalists.  " 
Nevertheless,  this  School  quibbled  and  argued  much  more  than  the 
masters  of  the  XlVth  Century,  and  lacked  their  originality. 

Joannes  Majoris,  who  was  primarily  a  great  teacher,  taught  Oresme's 
work  on  the  latitude  of  forms  and,  in  1504,  had  Buridan's  Summulae 
printed.  In  his  Disputationes  Theologiae  Majoris  argued,  more  explicitly 
than  Buridan  had  dared  to  do,  the  identity  of  the  dynamics  of  celestial 
and  terrestrial  bodies.  Thus,  like  Nicholas  of  Cues,  he  prepared  the  way 
for  Kepler.  But  it  was  left  to  him  to  fight  the  Reformation  and 
to  defend  the  dialectic  which  the  students  had  begun  to  neglect.  It  was 
a  time  when,  "  covered  with  threadbare  garments  and  with  empty 
purses,  the  unhappy  logicians  of  the  University  of  Paris  mused  sadly 
on  chairs  which  were  no  longer  surrounded  by  pupils.  They  listened  to 
the  raillery  that  was  poured  on  their  learning,  which  they  had  only 
acquired  with  great  effort,  and  to  which  they  had  consecrated  their 
working  lives.  " l  Already  attacked  by  the  Humanists,  Scholasticism 
no  longer  paid. 

In  1509  Jean  Dullaert  de  Gand  (1471-1513)  continued  the  printing 
of  Buridan's  works.  In  1506  he  himself  published  some  Quaestions  on 
Aristotle's  De  Caelo  and  Physics.  At  Montaigu  he  taught  the  doctrine 
of  impetus.  He  assumed  that  the  impetus  was  modified  by  the  shape 
of  the  projectile  and  supported  the  notion  of  intermediate  rest,  which 
he  took  to  be  at  the  moment  when  the  impetus  of  the  ascending  motion 
was  overcome  by  the  gravity.  He  came  to  no  conclusion  as  to  the 
nature  of  impetus,  whether  it  was  a  distinct  property  of  a  moving  body 
or  not.  Concerning  the  fall  of  heavy  bodies,  he  assumed  that  the  im 
petus  increased  continually,  though  he  did  not  know  whether  it  should 
be  taken  as  proportional  to  the  size  of  the  body.  Similarity,  Dullaert 
taught  Oresme's  rule  on  uniformly  deformed  motion  through  the  reading 
of  Bernard  Torni,  though  he  confined  the  treatment  to  the  algebraic 
form  of  the  rule.  He  lost  himself  in  discussions  on  the  nature  of  motion, 
a  "  successive  entity  truly  distinct  from  all  permanent  things.  " 

Luiz  Nunez  Coronel,  of  Segovia,  published  Physicae  Perscrutationes 
in  1511.  In  dynamics,  he  believed  in  the  gradual  weakening  of  the 
impetus  of  all  violent  motion,  which  was  a  serious  regression  from  Bu 
ridan's  thesis.  As  for  intermediate  rest,  he  "  imagined  instances  in 
which  a  stone  thrown  in  the  air  remains  there  at  rest  for  as  much  as  an 
hour,  two  hours  or  even  three,  "  without  being  dismayed  by  the  objection 
that  such  rest  was  never  seen.  "  This  objection  is  not  conclusive.  The 

1  DUHEM,  Etudes  sur  Leonard  de  Vinci,  Series  III,  p.  179. 


XVth   AND    XVIth    CENTURIES  89 

great  distances  may  prevent  one  from  seeing  the  rest,  or  it  may  even 
happen  that  the  stone  remains  motionless  for  a  time  which  is  imper 
ceptible.  " 

To  Coronel,  impetus  was  an  aptitude  of  the  moving  body,  a  certain 
"  actual  entity,  "  produced  in  it  by  means  of  a  repeated  series  of  local 
motions.  Impetus  was  thus  identified  with  a  cognition  acquired  by  the 
repetition  of  the  same  perception,  like  that  of  handwriting  to  the  fingers 
of  the  hand.  This  physiological  model,  however  arbitrary  it  may  be, 
was  taken  up  again  by  Kepler.  In  the  theory  of  gravitation,  Coronel 
showed  himself  to  be  singularity  naive.  If  weight  is  a  property  emanat 
ing  from  the  natural  place,  in  order  to  prevent  this  property  from  passing 
through  the  surface  of  the  earth,  it  will  be  sufficient  if  this  is  covered 
with  a  garment.  .  .  .  Elsewhere  Coronel  attributes  the  generation,  in 
a  free  fall,  of  an  impetus  of  greater  or  lesser  intensity,  exclusively  to 
gravitation  or  to  the  substantial  form  of  the  heavy  body. 

In  the  motion  of  projectiles,  Coronel  assumed  a  mixture  of  decreasing 
impetus  and  progressive  agitation  of  the  air,  which  resulted  in  a  certain 
compensation  and  assured  a  maximum  violence  at  the  middle  of  the 
trajectory. 

In  1517  Jean  de  Celaya  published  Expositio  in  libris  Physicorum,  a 
literal  commentary  on  Aristotle.  The  relevant  discussion  only  appear 
ed  later  under  the  title  Sequitur  glosa,  and  is  distinguished  by  having 
explained,  rather  clearly,  a  law  of  inertia  in  the  following  terms. 

"  It  would  follow  from  the  theory  that  a  body  which  is  projected 
will  move  forever.  However,  this  result  is  false  and  the  reason  is  clear. 
The  theory  does  not  include  anything  which  will  destroy  the  impetus, 
and  it  will  therefore  move  the  projectile  forever. 

u  To  this  we  reply  by  refusing  to  recognise  the  validity  of  the  argu 
ment,  and  this  because  we  deny  the  antecedent.  Indeed,  this  impetus 
is  sometimes  destroyed  by  the  resisting  medium,  sometimes  by  the 
shape  or  the  property  of  the  projectile  that  exerts  a  resisting  action, 
sometimes,  finally,  by  an  obstacle.  " 

Celaya  assumes  that  in  the  absence  of  these  three  mechanisms  of 
destruction,  impetus  lasts  indefinitely.  "  It  is  not  necessary  to  suppose 
as  many  intelligences  as  there  are  heavenly  bodies.  It  is  sufficient  to 
say  that  there  is  in  each  star  an  impetus?  that  this  impetus  was  put  there 
by  the  Prime  Cause,  and  that  it  is  this  which  moves  the  star.  This 
impetus  is  not  modified  for  the  very  reason  that  the  heavenly  body  has 
no  inclination  towards  a  different  motion*  " 

This  is  entirely  in  agreement  with  Buridan's  thesis. 

In  the  general  sense,  impetus  was  a  second  quality  to  Celaya.  He 
compared  it  to  **  knowledge  and  dispositions  of  the  soul.  ** 


90  THE    ORIGINS 

Celaya  was  rather  reticent  on  the  subject  of  the  plurality  of  worlds. 
The  Catholic  faith  provides  no  argument  from  which  the  existence  of 
several  worlds  can  be  deduced,  and  the  Philosopher  (Aristotle)  saw 
objections  to  such  a  happening.  All  the  same,  "  from  the  supernatural 
point  of  view,  there  can  exist  several  worlds,  either  simultaneously  or 
successively,  either  concentrically  or  excentrically. "  For  "  God  can 
do  all  things  that  do  not  imply  a  contradiction,  "  and  here  there  is  none. 
The  opinion  of  the  Philosopher  according  to  which  the  World  contains 
all  possible  matter  "  is  heretical,  and  the  Philosopher  would  not  be  able 
to  prove  it.  "  Finally,  we  remark  that  though  Celaya  taught  the  work 
of  Nicole  Oresme  and  the  Oxford  School,  like  Jean  Dullaert,  he  only 
knew  them  through  the  Italian  tradition. 


9.   THE  ATTACK  OF  THE  HUMANISTS. 

The  rapid  sketch  which  we  have  given  above  is  sufficient  to  show 
that  Parisian  Scholasticism  in  the  XVIth  Century  was  in  regression 
from  the  original  work  of  the  XlVth  Century.  The  Humanists  who 
were  to  proclaim  its  decadence  were  pupils  of  the  College  of  Montaigu — 
Didier  Erasme  and  Jean  Luiz  Vives. 

Moreover,  in  mechanics,  these  Humanists  preserved  the  tradition 
which  they  had  received  from  Majoris  and  Dullaert.  Thus,  in  his 
immensely  successful  Colloquia  (1522),  Erasme  discussed  the  oscillation 
of  a  heavy  body  that  travelled  through  to  the  centre  of  the  Earth — this 
is,  as  we  know,  a  problem  that  had  already  been  raised  by  Oresme — in 
terms  that  his  masters  would  not  have  disowned.  Erasme's  Eulogy  of 
madness,  which  antedates  the  Colloquia  and  was  published  in  1508, 
includes  a  determined  attack  on  the  theologians,  "  those  quibblers  who 
are  so  puffed  up  with  the  wind  and  smoke  of  their  empty  and  quite 
verbal  learning  that  they  will  not  give  way  on  any  point.  " 

Jean  Luiz  Vives  (1492-1540)  was  born  at  Valence  and  was  a  pupil 
of  Jean  Dullaert  before  becoming  a  professor  himself  at  Louvain.  In 
De  prima  philosophia  (1531)  he  discussed  "  intermediate  rest  "  at  great 
length,  in  terms  which  were  in  complete  conformity  with  the  pure  scho 
lastic  doctrine.  He  had  therefore  retained  traces  of  the  teaching  of 
Montaigu.  His  violent  diatribes  were  directed  at  the  Parisian  masters 
and  at  the  Oxford  School  with  its  XVth  Century  tradition,  which  sought 
to  extend  the  Calculator's  dialectic  to  medicine. 

In  De  philosophiae  naturae  corruptione  (1531)  Vives  wrote,  "  How  can 
there  be  learning  in  subjects  so  divorced,  so  completely  separated, 
from  God  on  the  one  hand  and  from  sensibility  and  spirit  on  the  other  ? 


XV th  AND    XVIth   CENTURIES  91 

In  a  domain  in  which,  founded  on  nothing,  there  is  seen  a  vast  structure 
of  contradictory  assertions  concerning  the  increase  and  decrease  of 
intensity,  the  dense  and  the  tenuous,  uniform  motion,  non-uniform 
motion,  uniformly  varying  motion  and  non-uniformly  varying  motion  ? 
It  is  not  possible  to  count  those  who,  without  any  limit,  discuss  instances 
which  never  occur,  which  could  never  turn  up  in  nature ;  who  talk  of 
infinitely  tenuous  and  infinitely  dense  bodies ;  who  divide  an  hour  into 
proportional  parts  for  this  reason  or  that,  and  consider,  in  each  of  these 
parts,  a  motion,  or  an  acceleration,  or  a  rarefaction,  varying  in  a  given 
way.  .  .  .  " 

Further,  in  De  medicina  we  find,  **  the  young  people  and  adolescents 
who  have  been  educated  by  means  of  these  specious  and  tricky  discus 
sions  know  nothing  of  plants,  of  animals,  nor  of  nature  in  the  round. 
They  have  been  brought  up  with  no  experience  of  natural  things, 
without  knowledge  of  reality.  They  have  no  prudence.  Their  judge 
ment  and  their  counsel  are  excessively  weak,  and  yet  they  are  expected 
to  be  able  to  win  honour  for  themselves  !  " 

And  he  concludes  ( In  pseudodialecticos) ,  "  For  myself,  I  have  a  great 
gratitude  to  God,  and  I  thank  him  that  I  have  at  last  left  Paris,  that 
I  have  emerged  from  the  Cimmerian  darkness,  have  come  out  into  the 
light,  that  I  have  discovered  the  truly  dignified  studies  of  mankind — 
those  which  have  earned  the  name  Humanities.  " 


10.  DOMINIC  DE  SOTO  (1494-1560)  AND  THE  LAWS  OF  FALLING  BODIES. 

At  the  very  moment  that  Scholasticism  appeared  to  be  discredited 
by  the  attacks  of  the  Humanists,  there  intervened  an  original  work 
which  succeeded  in  formulating  the  laws  of  falling  bodies  correctly.  We 
shall  now  analyse  this  work  in  some  detail. 

Dominic  de  Soto  was  born  in  1494,  the  son  of  a  gardener  at  Segovia. 
He  attended  the  University  of  Alcala  of  Henares,  and  then  took  himself 
to  the  University  of  Paris  where  the  Spaniards  were  already  rather 
numerous.  He  returned  to  Alcala  in  1520  and  gave  up  the  chair  which 
he  had  obtained  in  order  to  take  the  habit  of  a  preaching  friar.  From 
1532  to  1548  he  taught  theology  at  Salamanca.  As  confessor  to  Charles  V, 
he  followed  his  king  to  Germany.  Later  he  returned  to  Salamanca 
and  taught  theology  there  from.  1550  until  his  death  in  1560. 

Soto  had  been  a  witness  of  the  furious  attacks  of  the  Humanists  upon 
the  Paris  School  but  remained,  for  his  part,  a  Schoolman.  However, 
he  eschewed  nominalism  and  attacked  it  in  his  Quaestiones  (1545)  on 
Aristotle's  Physics. 


92  THE   ORIGINS 

We  shall  not  discuss  the  metaphysical  content  of  Soto's  work,  in 
which  he  rejected  the  concept  of  an  actual  infinity  in  favour  of  a  virtual 
one,  and  shall  only  be  concerned  with  his  contribution  to  mechanics. 
In  the  first  place  we  note,  in  passing,  that  Soto  adopted  Albert  of 
Saxony's  opinion  on  the  equilibrium  of  the  earth  and  the  seas.  In 
connection  with  the  motion  of  projectiles,  he  taught  the  doctrine  of 
impetus,  and  presented  it  in  the  following  way. 

"  First  Conclusion.  —  It  cannot  be  denied  that  a  man  or  a  mechan 
ism  sets  the  air  in  motion  when  throwing  a  projectile,  just  as  we  see 
the  circular  agitation  of  water  around  a  stone  which  has  been  thrown 
into  it.  The  truth  of  this  conclusion  is  especially  evident  for  cannons, 
from  which  the  air  is  driven  in  the  form  of  a  very  violent  explosion  at 
the  same  time  as  the  shot. 

"  Second  Conclusion.  —  Air  is  not  the  only  cause  of  the  motion  of 
projectiles.  Whatever  has  thrown  the  moving  body  is  also  a  cause, 
through  the  intermediary  of  the  impetus  which  it  has  impressed  on  the 
body.  "  i 

Thus  Soto  sought  to  reconcile  Aristotle's  doctrine  with  that  of 
impetus  by  assuming  that  the  agitation  of  the  air  played  some  part  in 
the  motion  of  projectiles.  However,  he  summarily  dismissed  Marsile 
Inghen's  opinion  (see  above,  page  68). 

"  Observation  proves  that  air  too  is  a  cause  of  the  motion  of  pro 
jectiles.  Indeed,  we  know  that  an  arrow  does  not  hit  an  object  which 
is  near  with  as  much  violence  as  it  hits  one  that  is  a  little  more  distant. 
This  is  why  Aristotle  says,  in  the  second  book  of  the  Heavens,  that 
natural  motion  is  more  intense  towards  the  end,  while  the  greatest 
intensity  of  the  motion  of  a  projectile  is  attained  neither  at  the  beginn 
ing  nor  at  the  end,  but  near  the  centre. 

"  Some  suppose  that  the  reason  for  this  happening  is  the  following 
one —  the  impetus  is  not  all  imparted  to  the  arrow  at  the  first  instant. 
Later  it  becomes  more  intense,  or  else  distributed  through  the  extension 
of  the  arrow,  so  that  it  moves  it  in  a  more  urgent  way.  But  this  is  not 
very  easy  to  understand.  Indeed,  one  cannot  see  what  could  increase 
the  intensity  of  the  impetus  after  the  arrow  has  been  separated  from  the 
ballista,  for  an  accident  does  not,  of  itself,  become  more  intense.  On 
the  other  hand,  as  the  arrow  is  a  continuous  body,  the  impetus  is  simul 
taneously  imparted  to  the  whole  body.  Therefore  it  cannot  distribute 
itself  further  later.  "  2 

1  Quaestiones  in  libros  Physicorum,  Vol.  II,  fol.  100. 

2  Ibid. 


XVth   AND    XVIth   CENTURIES  93 

Soto  regarded  impetus  as  a  "  property  distinct  from,  the  subject  in 
which,  it  is  encountered,  "  like  gravity  or  lightness.  Conversely,  he 
saw  gravity  as  a  "  natural  impetus.  " 

In  his  desire  to  reconcile  Aristotle  and  Buridan,  Soto  went  as  far  as 
to  argue  that  Aristotle  did  not  doubt  the  doctrine  of  impetus,  but  that 
he  must  have  taken  it  as  obvious,  from  the  analogy  with  heavy  and 
light  bodies,  and  passed  over  it  in  silence. 

But  the  essential  part  of  Soto's  work  is  that  which  concerns  the  fall 
of  bodies.  Some  of  the  Schoolmen  who  had  proceeded  him  had  discussed 
the  fall  of  bodies,  albeit  in  a  purely  qualitative  manner;  others  had 
discussed  uniformly  varying  motion  in  the  field  of  pure  kinematics ; 
but  these  studies  had  remained  separate.  It  has  now  been  established 
that  the  synthesis  of  these  discussions  was  accomplished  in  Soto's  time. 
He  himself  does  not  describe  this  achievement  as  a  personal  success.  Is 
this  modesty  on  his  part  or,  on  the  other  hand,  the  reflection  of  a  move 
ment  which  had  already  been  completed  by  the  Schoolmen  ?  The 
answer  to  this  question  is  of  little  importance — what  does  matter  is  the 
law  which  was  clearly  expressed  by  this  Spanish  master. 

We  shall  quote  Soto's  own  text,  as  translated  by  Duhem.1 

"  Motion  which  is  uniformly  deformed  with  respect  to  time  is  that 
in  which  the  deformity  is  so —  if  it  is  divided  according  to  time,  that  is 
according  to  intervals  which  succeed  each  other  in  time,  in  each  part 
the  motion  at  the  central  point  exceeds  the  weaker  terminal  motion  in 
this  part  by  an  amount  equal  to  that  by  which  it  itself  exceeded  by  the 
more  intense  terminal  motion, 

"  This  kind  of  motion  is  one  which  is  appropriate  to  bodies  which 
have  a  natural  motion  and  to  projectiles  (Haec  motus  species  proprie 
accidit  naturaliter  motis  et  projectis) . 

u  Indeed,  each  time  that  a  mass  falls  from  the  same  height  in  a 
homogeneous  medium,  it  moves  more  quickly  at  the  end  than  at  the 
beginning.  On  the  contrary,  the  motion  of  bodies  which  are  projected 
[upwards]  is  weaker  at  the  end  than  at  the  beginning.  And  similarily 
the  first  motion  is  uniformly  accelerated  and  the  second,  uniformly 
retarded.  " 

Soto  was  concerned  with  the  law  of  distances  for  uniformly  varying 
motion,  and  in  his  writings  the  ideas  of  Nicole  Oresme  and  of  the  Oxford 
School  may  be  clearly  identified.  After  some  hesitation  he  declared 
himself  for  the  correct  law. 

"  Uniformly  deformed  motion  with  respect  to  time  follows  almost 
the  same  law  as  uniform  motion  does.  If  two  bodies  travel  equal 

1  Theologi  ordinis  pr dedicator um  super  octo  libri  Physicorum  Aristotelis  Quaestiones, 
Salamanca,  1572,  fol.  92  d. 


94  THE   ORIGINS 

distances  in  a  given  time,  even  though  one  moves  uniformly  and  the 
other  in  any  deformed  manner — for  example,  in  such  a  way  that  it 
covers  one  foot  in  the  first  half-hour  and  two  feet  during  the  second — 
from  the  moment  that  the  latter  covers  as  many  feet  as  the  former, 
which  moves  uniformly,  in  the  whole  hour,  the  two  moving  bodies 
will  move  equally. 

"  But  here  an  uncertainty  arises.  Should  the  velocity  of  a  body 
in  uniformly  varying  motion  be  denominated  by  its  most  intense  degree  ? 
If  for  example,  the  velocity  of  a  falling  body  increases  in  one  hour 
from  degree  zero  to  degree  eight,  should  it  be  said  that  this  body  has 
a  motion  of  degree  eight  ?  It  seems  that  the  affirmative  reply  is  the 
correct  one,  for  this  is  the  law  which  seems  to  be  followed  by  uniformly 
varying  motion  with  respect  to  a  subject  moving  body.  Nonetheless 
we  reply  that  the  velocity  of  uniformly  varying  motion  is  evaluated 
by  the  mean  degree  and  should  be  given  the  denomination  of  that 
degree.  One  should  not  argue  in  this  respect  as  in  the  case  of  uniformly 
varying  motion  with  respect  to  the  subject.  Indeed,  in  the  latter 
case  the  reason  for  the  rule  adopted  is  the  following —  each  part  of 
the  moving  body  describes  the  same  line  as  the  most  rapidly  moving 
point,  in  such  a  way  that  the  whole  moves  as  quickly  as  this  point. 
Whereas  a  body  which  moves  with  a  motion  that  is  uniformly  deformed 
with  respect  to  time  does  not  describe  a  path  as  great  as  if  it  were  mov 
ing  uniformly  with  the  velocity  which  it  attains  at  its  supreme  degree. 
This  goes  without  saying.  Therefore  we  believe  that  uniformly  de 
formed  motion  should  be  denominated  by  its  mean  degree.  Example — 
//  the  moving  body  A  moves  for  one  hour  and  constantly  accelerates  its 
motion  from  degree  zero  to  degree  eight,  it  will  travel  just  as  great  a  path 
as  the  moving  body  B  which  moves  uniformly  with  degree  four  for  the  same 
time. 

"  It  follows  from  this  that  when  bodies  move  with  a  deformed 
motion,  these  motions  should  be  reduced  to  uniform  ones.  "  1 

1  Ibid.,  fol.  93  and  94. 


CHAPTER   SIX 

XVIth  CENTURY 
(Continued) 

THE  ITALIAN  SCHOOL 
OF  NICHOLAS  TARTAGLIA  AND  BERNARDINO  BALDI 


1.  NICHOLAS  TARTAGLIA. 

Nicholas  Fontana,  called  Tartaglia,  derived  his  surname — which 
indicates  stammering — from  an  injury  obtained  when  he  was  wounded, 
while  still  an  infant,  in  the  sack  of  Brescia.  He  was  born  at  Brescia 
at  the  beginning  of  the  XVIth  Century  and  died  at  Venice  in  1557. 

Tartaglia  was  one  of  the  means  by  which  the  original  statics  of 
the  XHIth  Century,  which  had  been  forgotten,  was  preserved  for  the 
Italian  School  of  the  XVIth  Century.  Indeed,  Tartaglia  entrusted 
Curtius  Trojanus  with  the  publishing  of  the  work  of  the  unknown 
author  of  the  XIHth  Century.  This  appeared  in  1565  under  the 
title  of  Jordani  opusculum  de  ponderosite  Nicolai  Tartaleae  studio 
correctum.1  Instead  of  giving  his  predecessors  credit  for  their  work, 
Tartaglia,  who  was  not  very  scrupulous  in  matters  of  scientific  pro 
priety,  claimed  their  demonstrations  as  his  own. 

Dynamics  is  treated  in  two  of  Tartaglia's  works,  Nova  Scientia 
(1537)  and  Quesiti  et  inventioni  diversi  (1546). 

In  the  first  of  these  works  Tartaglia  attributes  the  acceleration 
of  falling  bodies  to  their  approach  to  their  natural  place.  "  A  heavy 
body  hastens  towards  its  proper  nest,  which  is  the  centre  of  the  World, 
and  if  it  comes  from  a  place  which  is  more 
distant  from  this  centre  it  will  travel  more 
quickly  in  approaching  it.  " 

Elsewhere  he  distinguished  three  phases 
in  the  trajectory  of  a  projectile —  AB 
(rectilinear),  BC  (a  curved  join)  and  CD 
(vertical).  He  held  that  the  velocity  was  Fig.  27 

1  See  above,  pp.  41  to  46. 


96  THE    ORIGINS 

least  at  C,  at  the  point  at  which  the  violent  motion  finished  and  the 
natural  motion  began. 

This  dynamics  is  improved  a  little  in  the  Quesiti,  in  which  he 
asserts  that,  except  for  the  case  in  which  the  particle  is  thrown 
vertically,  the  trajectory  of  a  shot  has  no  rectilinear  portion.  It  is 
the  natural  gravity  which  makes  the  trajectory  curve  downwards. 
The  more  rapidly  a  heavy  body  is  thrown  in  the  air,  the  less  heavy  it 
is  and  the  straighter  it  travels  through  the  air,  which  supports  a 
lighter  body  more  effectively.  The  more  the  velocity  decreases  the 
more  the  gravity  increases,  and  this  gravity  continually  acts  upon  the 
body  and  draws  it  towards  the  earth.1 

Tartaglia  adds  that  the  motion  of  a  projectile  starts  with  an  accel 
eration.  He  says  that,  for  the  same  cannon  with  the  same  charge 
of  powder  and  the  same  elevation,  a  second  shot  will  go  further  than 
the  first  because  it  will  find  the  air  already  divided  and  more  easily 
penetrable. 

2.  JEROME  CARDAN  (1501-1576). 

Jerome  Cardan  was  born  at  Padua  in  1501,  died  at  Rome  in  1576, 
and  was  at  once  physician,  astrologer,  algebraist  and  a  student  of 
mechanics. 

Cardan's  two  works  which  are  relevant  to  mechanics  are  the  De 
Subtilitate  (1551) — which  was  translated  into  French  by  Richard  le 
Blanc  in  1556 — and  the  Opus  novum  (1570). 

In  statics  Cardan  believed  that  that  he  had  surpassed  Archimedes, 
whom  he  had  read  and  admired,  by  treating  the  weight  of  the  two 
arms  of  a  balance.  Indeed  he  wrote,  "  The  heavinesses  [moments]  of 
the  two  arms  of  a  beam  [horizontal,  cylindrical  and  homogeneous] 
have  the  same  proportion  to  each  other  as  that  of  the  squares  of  the 

lengths   of  the   two   arms  .  .  .  Hoc   est   quod 
Archimedes  reliquit  intactum.  "  2 

Cardan  uses  the  concept  of  moment  fully. 
"  It    is    clear   that,    in    balances    and    in 
things  which  lift  loads,  the  further  the  burden 
is  from  the  fulcrum  the  heavier  it  is.     Now  the 
weight  at   C  is  separated  from  the  fulcrum 

by  the  length  of  the  line  CJ5  and  that  at  JP, 

Fig.  28  by  the  length  of  the  line  FP.  "  3 

1  C/.  DUHEM,  Etudes  sur  Leonard  de  Vinci,  Series  III,  p.  188.     It   may  be  that 
Tartaglia  used  Leonardo  da  Vinci's  notes  without  acknowledgement. 

Opus  novum,  Proposition  XCIL 
3  De  subtilitate,  translated  by  RICHARD  LE  BLANC,  p.  16. 


XVIth    CENTURY  97 

Like  his  predecessors  of^the  Xlllth  Century,  Cardan  then  consi 
dered  equal  arcs  FG  and  CE  starting  from  the  points  jP  and  C,  but 
he  directed  his  attention  to  the  velocities  and  not  to  the  paths,  by 
observing  that  the  fall  from  F  to  G  was  more  "  tardy  "  than  the  fall 
from  C  to  E. 

He  concludes,  "  then  this  argument  is  general — that  the  further 
the  weights  are  from  the  end,  or  the  line  fall  of  along  the  straight  line 
or  the  oblique,  that  is  to  say  along  the  angle,  the  heavier  they  are.  .  .  . 
Thus  the  intention  of  the  weight  is  to  be  carried  directly  towards  the 
centre.  But  because  it  is  prevented  from  doing  this  by  the  linkages, 
it  moves  as  best  it  can.  "  l 

Duhem  interprets  this  rather  obscure  passage  in  the  following  way. 

"  When  a  heavy  body  falls  vertically  the  power  of  the  body  is 
measured,  as  Aristotle  intended,  by  the  velocity  with  which  it  falls. 
But  through  the  agency  of  the  mechanism  that  carries  it,  because 
of  the  linkages  or  constraints — to  use  the  modern  term — it  may  happen 
that  the  body  does  not  move  vertically.  Therefore  in  order  to  reckon 
its  motive  power  it  is  necessary  to  take  account,  not  of  the  body's 
total  velocity,  but  only  of  the  vertical  component  of  this  velocity, 
or  in  other  words,  of  the  velocity  of  fall. 

"  If  then  a  given  weight  is  suspended  from  some  point  of  a  solid 
which  can  move  about  a  horizontal  axis,  the  power  of  this  weight 
will  be  greater  as  the  point  of  suspension  falls  more  rapidly  when  a 
given  rotation  is  applied  to  the  support.  Therefore  it  will  be  greater 
as  the  point  of  suspension  is  further  from  the  vertical  plane  containing 
the  axis.  "  2 

Like  Leonardo,  Cardan  investigated  the  pul 
ley-block,  together  with  the  screw  and  the  jack. 

Cardan  was  of  the  opinion  that  on  an 
inclined  plane  the  heaviness  of  a  given  body 
is  proportional  to  the  velocity  with  which  it 
moves  down  the  plane.  Therefore  this  hea 
viness  is  zero  on  a  horizontal  plane  and 
increases  with  the  angle  of  inclination.  Car 
dan  assumes  that  the  apparent  weight  is  (, 
proportional  to  this  angle. 

"  Let  a  sphere  a,  of  weight  £,  be  placed  at  <lg"  "" 

the  point  b  and  suppose  that  it  is  desired  to 

draw  it  along  the  plane  be.     The  vertical  plane  is  bf.     On  the  horizontal 
plane  be  the  force  needed  to  move  a  may  be  taken  as  small  as  desired. . . . 

1  Ibid. 

2  DUHEM,  0.  S.,  Vol.  I,  p.  46. 


98  THE    ORIGINS 

Consequently,  according  to  tke  consensus  of  opinion,  the  force  which 
will  move  a  along  be  will  be  zero.  On  the  other  hand,  a  will  be  moved 
towards  /  by  a  constant  force  equal  to  g  ;  in  the  direction  be  by  a 
constant  force  equal  to  fc ;  in  the  direction  bd  by  a  constant  force  equal 
to  ft.  Since  the  motion  along  be  is  produced  by  a  zero  force,  the  rela 
tion  of  g  to  k  will  be  as  the  relation  of  the  force  which  moves  a  along  bf 
to  the  jforce  which  moves  a  along  6c,  and  as  the  relation  of  the  right 
angle  ebf  to  the  angle  ebc.  In  the  same  way  the  force  which  moves  a 
along  bf  isjx>  the  force  which  moves  a  along  bd  as  the  angle  ebf  is  to 
the  angle  ebd.  "  l 

There  is  no  clearer  distinction  between  statics  and  dynamics  in 
Cardan's  work  than  can  be  discovered  in  that  of  Leonardo  da  Vinci. 
Like  Leonardo,  Cardan  asserted  the  impossibility  of  perpetual  motion 
unless  natural  motions  were  in  question.  We  shall  quote  De  Subtilitate 
on  this  subject. 

"  Either  the  continuity  of  motion  will  arise  from  the  fact  that  the 
motion  is  in  conformity  with  nature,  "  (hereby  Cardan  excepts  the 
motion  of  the  Heavens),  "  or  else  this  continuity  will  not  be 
maintained  equal  to  itself.  Now  that  which  continually  diminishes 
and  is  not  augmented  by  some  external  action,  cannot  be  per 
petual.  .  .  . 

"  The  motions  that  bodies  can  have  are  of  three  kinds  ;  they  may 
essentially  tend  to  the  centre  of  the  World  ;  they  may  not  be  directed 
towards  the  centre  in  a  simple  way,  like  the  running  of  water  ;  or  they 
may  stem  from  a  particular  characteristic,  like  the  motion  of  iron  to 
wards  a  magnet.  Patently,  perpetual  motion  should  be  sought  in  mo 
tions  of  the  first  two  kinds.  Now  when  a  weight  is  pulled  more  strongly, 
or  held  back  more  energetically,  than  is  consistent  with  its  nature  its 
motion  is  natural,  it  is  true,  but  not  free  of  violence.  Examples  of  these 
two  conditions  are  seen  in  the  weights  of  clocks.  As  for  motion  in  a 
circle,  this  only  belongs  naturally  to  the  sky  and  the  air,  and  the  latter 
is  not  actuated  by  an  ever-present  mechanism.  For  other  bodies,  it 
[motion  in  a  circle]  always  has  its  root  in  vertical  motion.  Thus  in 
rivers,  at  the  rate  and  to  the  extent  that  the  waters  are  generated  by 
the  source,  they  continually  descend  along  the  slope  of  the  bed.  Now 
in  order  that  a  motion  should  be  perpetual,  it  would  be  necessary 
that  the  bodies  which  were  displaced  and  came  to  the  end  of 
their  path  should  be  carried  back  to  their  initial  position.  But  they 
can  only  be  carried  there  by  means  of  a  certain  excess  of  motive 
power.  ..." 

1  Opus  novum.  Proposition  LXXIL 


XVIth    CENTURY  99 

3.  JULIUS-CAESAR  SCALIGER  AND  BURIDAN'S  DOCTRINE. 

Julius-Caesar  Scaliger  was  a  supporter  of  the  parisian  Scholasticism 
and  one  of  Cardan's  opponents.  The  latter,  in  Book  XVI  of  De  Subti- 
litate,  had  had  the  naive  audacity  to  make  a  classification  of  genius,  in 
order  of  decreasing  merit,  in  the  following  way —  Archimedes,  Aristotle, 
Euclid,  John  Duns  Scot,  Swineshead  the  Calculator,  Apollonius  of 
Pergum,  Archytas  of  Tarento,  etc.  .  .  .  Scaliger  replied  on  this  matter. 
"  You  have  given  a  simple  artisan  the  place  above  Aristotle,  who  was 
not  less  erudite  than  he  in  these  same  mechanical  skills  ;  above  John 
Duns  Scot,  who  was  like  the  file  of  truth  ;  above  Swineshead  the  Calcu 
lator,  who  almost  surpassed  the  limits  imposed  on  the  human  intelli 
gence  !  You  have  passed  over  Ockham  in  silence,  that  genius  who 
outwitted  all  previous  geniuses.  .  .  .  You  have  placed  Euclid  after 
Archimedes,  the  torch  after  the  lantern.  .  .  .  "  x 

Scaliger  explicitly  refused  to  consider  the  agitated  air  as  the  seat 
of  the  motive  agency  of  projectiles,  and  accepted  Buridan's  doctrine  in 
all  but  form.  In  this  he  differed  from  Cardan,  who  remained  an  Aristo 
telian  in  this  matter  and  who  added  nothing  to  the  work  of  Tartaglia 
and  da  Vinci. 

"  The  motio  (here  synonymous  with  impetus)  is  an  entity  which 
implanted  in  the  moving  body  and  which  can  remain  there  even  when 
the  prime  mover  is  taken  away.  By  prime  mover  I  mean  that  which 
causes  this  entity  to  penetrate  into  the  body.  For  it  is  not  necessary 
that  the  efficient  cause  should  continue  to  exist  with  its  effect.  "  2 

Scaliger  continued — 

"  Heavy  bodies,  stones  for  example,  have  nothing  which  favours 
their  being  set  in  motion.  They  are,  on  the  contrary,  quite  opposed  to 
it.  ...  Why  then  does  a  stone  move  more  easily  after  the  motion  has 
started  ?  Because  the  stone  has  already  received  the  impression  of 
motion.  To  a  first  part  of  the  motion  a  second  succeeds,  and  each  time 
the  first  remains.  So  that,  rather  than  a  single  motor  exerting  its 
action,  the  motions  which  it  imparts  in  this  continuous  succession  are 
multiplied.  For  the  first  impetus  is  kept  by  the  second,  and  the  second 
by  the  third "  3 

4.  BENTO  PEREIRA  (1535-1610).    THE  CLASSICAL  REACTION. 

In  1562  Bento  Pereira  published  at  Rome  a  treatise  called  De  com- 
munibus  omnium  rerum  naturalium  principiis  which  became  very  po- 

1  Exotericarum  evercitationum  libri,  Paris  1 557t  Exerc.  324.    Translated  by  DUHEM. 

2  Ibid.,  Exerc.  76. 

3  Ibid.,  Exerc.  77. 


100  THE    ORIGINS 

pillar  and  which  was  studied  by  Galileo  himself.  Bento  Pereira  knew 
of  Scaliger's  Exercitationes  but  adhered,  himself,  to  Aristotle's  doctrine 
on  the  motion  of  projectiles.  Cesalpin  and  Borro  may  also  be  cited  as 
representatives  of  this  classical  reaction. 

5.  THE  "  MECHANICORUM  LIBER  "  OF  GUIDO  UBALDO  (1545-1607). 

We  now  come  to  a  student  of  mechanics  who  was  a  great  authority 
until  the  beginning  of  the  XVIIIth  Century,  and  who  was  one  of  Galileo's 
masters.  Descartes,  who  gave  few  references,  recalled  having  read  him 
and  even  Lagrange  quoted  him  often  in  the  historical  part  of  his  Meca- 
nique  Analytique.  To  the  classical  impedimenta  of  the  Medieval  authors, 
Guido  Ubaldo  added  a  reading  of  Archimedes  and  of  Pappus,  and 
through  the  latter  achieved  a  partial  knowledge  of  Hero  of  Alexandria. 
His  Mechanicorum  Liber  is  dated  1577. 

Guido  Ubaldo,  who  was  Marquis  del  Monte,  lived  in  seclusion  in  his 
Castle  del  Monte  Barrochio,  and  devoted  all  his  leisure  to  study. 

In  his  writings  on  statics  he  reproached  the  Schoolmen  of  the  Xlllth 
Century,  with  good  reason,  for  having  made  a  first  principle  of  gravitas 
secundum  situm  without  having  justified  this  action  in  any  way.  He 
wished  to  see  substituted  for  this  concept,  the  effect  of  the  reaction  of 
the  support. 

"  The  mind  cannot  be  at  peace  while  the  variation  of  gravity  has  not 
been  attributed  to  some  other  cause  than  this.  Indeed,  it  seems  that 
[the  variation  secundum  situm]  is  a  symbol  rather  than  a  true  reason. 

"  The  line  CD  resists  a  weight  placed  at 
D  less  than  the  line  CL  resists  a  weight  at  L. 
Thus,  then,  the  same  weight  can  be  heavier 
or  lighter  in  virtue  of  the  effect  of  the  posi 
tion  it  occupies  ;  not  that  by  the  very  fact 
of  this  situation  it  really  acquires  a  new 
gravity  or  that  it  loses  its  original  gravity — 
rather  it  always  keeps  the  same  gravity  in 
whatever  place  it  may  be  ;  but  because  it 
always  weighs  more  or  less  on  the  circumfer- 
ence.  " 

Guido  Ubaldo  confined  himself  to  this 

qualitative  statement,  for  he  did  not  have  at  his  disposal  the  law  of 
the  composition  of  forces. 

Nevertheless,  he  used  the  concept  of  moment  to  substantiate  the 
condition  for  the  equilibrium  of  a  lever,  by  means  of  an  argument  whose 
form  is  directly  inspired  by  Archimedes.  He  corrected  certain  errors  in 


XVIth   CENTURY  101 

the  Xlllth  Century  discussion  of  the  stability  of  the  balance,  but  made 
the  mistake  of  using  the  same  treatment  when  the  verticals  were  assum 
ed  parallel  as  he  used  when  they  were  supposed  to  converge. 

Guido  Ubaldo  favoured  Pappus'  solution  of  the  problem  of  the 
inclined  plane — we  have  already  seen  the  weakness  and  superficial 
character  of  this  solution.  Thus  he  was  led  to  attribute  a  gravity  to  a 
moving  body  situated  on  a  horizontal  plane,  contrary  to  the  content  of 
Xlllth  Century  statics.  However,  he  in  general  preferred  to  consider 
virtual  displacements  than  virtual  velocities.  He  said  that  it  is  necessary 
to  deploy  a  greater  power  in  order  to  move  a  body  than  is  necessary  to 
maintain  it  in  equilibrium,  which  shows  that  he  did  not  understand  the 
part  played  by  the  passive  resistances. 

Guido  Ubaldo  took  over  Pappus'  definition  of  the  centre  of  gravity 
and  supplemented  it  with  the  following  commentary,  which  was  to  have 
a  great  influence  on  the  authors  of  the  XVIIth  Century. 

"  The  rectilinear  fall  of  bodies  shows  clearly  that  heavy  bodies  fall 
according  to  their  centres  of  gravity.  .  .  .  Strictly  speaking,  a  heavy 
body  weighs  through  its  centre  of  gravity.  The  very  name  centre  of 
gravity  seems  to  declare  this  truth.  Clearly,  all  the  force,  all  the  gravity 
of  the  weight  is  massed  and  united  at  the  centre  of  gravity  ;  it  seems  to 
run  from  all  sides  towards  this  point.  Because  of  its  gravity,  indeed,  the 
weight  has  a  natural  desire  to  pass  through  the  centre  of  the  Universe. 
But  it  is  the  centre  of  gravity  that  properly  tends  to  the  centre  of  the 
World.  " 

Thus  to  Guido  Ubaldo,  just  as  much  as  to  the  writers  of  the  XlVth 
Century,  the  concept  of  centre  of  gravity  was  a  purely  experimental  one. 
It  was  not  linked  in  any  way  with  the  parallelism  of  verticals. 

Guido  Ubaldo's  works,  "  sometimes  erroneous,  always  mediocre, 
were  often  a  regression  from  the  ideas  that  had  inspired  the  writings  of 
Tartaglia  and  Cardan.  "  T  However,  this  work  is  a  milestone  in  the 
history  of  mechanics  in  that  it  had  a  direct  stimulating  influence  on  the 
great  founders  of  mechanics,  to  whom  it  brought  the  content  of  the 
researches  of  Antiquity  and  the  Middle  Ages.  Its  value  was  at  least 
that  of  a  link  with  the  past. 

6.   J.-B.  VlLLALPAND  (1552-1608)  AND  THE  POLYGON  OF  SUSTENTATION. 

J.-B.  Villalpand  was  born  at  Cordoba  in  1552  and  belonged  to  the 
Society  of  Jesuits.  He  became  concerned  with  mechanics  because  of 
the  diversion  of  an  archeological  mission  to  Jerusalem.  He  took  it 

1  DUHEM,  0.  S.,  Vol.  I,  p.  226. 


102  THE    ORIGINS 

upon  himself  to  refute  certain  of  EzechiePs  commentators,  who  had 
claimed  that,  because  of  its  physical  geography,  Judea  offered  better 
possibilities  for  agriculture  and  construction  than  a  plain  of  the  same 
area  would  have  done. 

This  explains  the  title  of  Villalpand's  book,  Apparatus  Urbis  ac 
Templi  Hierosolymitani,  which  was  printed  at  Rome  in  1603. 

In  it  Villalpand  states,  among  others,  the  following  proposition — 
"  A  heavy  body  that  rests  on  the  ground  and  covers  a  certain  area 
remains  in  equilibrium  when  the  vertical  drawn  through  the  centre  of 
this  area  passes  through  the  centre  of  gravity  ;  or,  otherwise,  when  a 
vertical  drawn  through  the  edge  of  this  area  passes  through  the  centre 
of  gravity  or  leaves  it  on  the  same  side  as  the  area.  But  if  it  leaves  the 
centre  of  gravity  on  the  other  side  of  the  area,  the  heavy  body  will 
necessarily  fall. " 

Here  is  his  proof — 

"  If  the  line  FC,  when  produced,  leaves  the  centre  of  gravity  L  of 
the  body  on  the  opposite  side  to  the  area  BC  upon  which  the  heavy 
body  rests,  the  body  will  necessarily  fall.  Indeed,  the  weight  CLG  is 


H 


Fig.  31 


equal  to  the  weight  CLA.  The  weight  CGH  will  be  greater  than  the 
weight  CHA.  The  heavier  volume  will  drag  the  less  heavy  one  along. . .  , 
and  the  body  will  fall  on  the  side  of  G.  " 

It  is  quite  probable  that  Villalpand,  either  directly  or  otherwise, 
borrowed  this  result,  together  with  his  later  considerations  on  the  walk- 


XVIth    CENTURY 


103 


ing  of  living  beings  and  the  flight  of  birds,  from  da  Vinci.  However  it 
may  be,  we  are  indebted  to  P.  Mersenne  for  having  made  the  preceding 
theorem  on  the  polygon  of  sustentation  classical.  That  tireless  scholar 
was  able  to  extract  it  from  the  religious  exposition  in  which  it  had  been 
lost,  and  to  reproduce  it  in  his  collection  Synopsis  mathematica  (Mecha- 
nicorum  libri),  published  at  Paris  in  1626. 


f 


7.  J.-B.  BENEDETTI  (1530-1590).     STATICS.     FIGURE  OF  THE  EARTH. 
DOCTRINE  OF  "  IMPETUS.  " 

From  the  start  of  his  scientific  career  in  1553  Benedetti  denied  the 
truth  of  the  following  proposition  of  Aristotle,  a  proposition  which  had 
been  adopted  by  Jordanus's  School —  Let  two  bodies,  A  and  J5,  be  made 
of  the  same  substance  and  let  A  have  twice  the  volume  of  B.  The 
velocity  of  fall  for  A  is  twice  that  for  B. 

More  generally,  Benedetti  rejected  Aristotle's  statics.  "  The  laws 
of  the  lever,  "  he  wrote,  "  do  not  depend  in  any  way  on  the  rapidity  or 
on  the  extent  of  the  motion.  "  This  does  not  mean  that  he  adopted 
Jordanus's  doctrine,  or  in  other  words,  that  he  substituted  the  concept 
of  virtual  work  for  that  of  virtual  velocities.  In  fact,  he  reduced  the 
whole  of  statics  to  the  single  rule  of  the  lever  and  the  concept  of  moment. 

"  The  ratio  of  the  gravity  of  the  weight 
placed  at  C  to  the  gravity  of  the  weight  placed 
at  F  is  equal  to  the  ratio  of  BC  to  Bu.  .  „  . 
This  will  appear  evident  to  us  if  we  imagine  a 
vertical  thread  jFu,  and  if  we  imagine  that  the 
weight  at  jF  hangs  from  the  end  of  the  thread 
at  M.  It  is  clear  that  the  weight  hung  in  this 
way  would  produce  the  same  effect  if  it  were 
placed  at  F.  "  It  seems  that  Benedetti  had 
an  inkling  of  the  general  utilisation  of  mo 
ments  for  measuring  the  effects  of  weights  or 
of  any  motive  powers  whatever. 

To  a  certain  extent  then  this  criticism  of  Benedetti's  was  useful  and 
constructive.  On  the  other  hand  his  rejection  of  the  solution  of  the 
problem  of  the  inclined  plane,  due  to  the  unknown  author  of  the  Xlllth 
Century,  and  his  repetition  of  Leonardo  da  Vinci's  errors  concerning  the 
division  of  a  weight  between  two  convergent  supports,  was  less  fortunate. 

In  the  matter  of  the  figure  of  the  Earth  and  the  separation  between 
the  continents  and  the  oceans,  Benedetti  found  his  inspiration  in  Coper 
nicus.  In  1579  he  denied  the  truth  of  Albert  of  Saxony's  opinions  in 
the  following  terms. 


M 


Fig.  32 


104  THE    ORIGINS 

"  We  are  certain  that  the  spherical  surface  of  the  water  is  everywhere 
equidistant  from  the  centre  of  the  Universe,  the  point  sought  by  all 
heavy  bodies.  Moreover,  because  of  the  numerous  islands,  because  of 
the  different  countries  which  navigation  has  discovered  in  all  regions, 
we  can  be  sure  and  certain  that  the  water  and  the  earth  comprise  one 
globe,  and  that  the  geometrical  centre  of  the  Earth,  together  with  the 
centre  of  its  gravitation,  is  at  the  centre  of  the  Universe.  " 

We  must  add  that  Benedetti  considered  the  copernican  system  to  be 
a  plausible  one,  though  he  did  not  accept  it  himself. 

It  is  said  that  Benedetti's  works,  united  under  the  title  Diversarum 
speculationum  mathematicarum  et  physicarum  and  published  in  1585, 
covered  all  the  branches  of  mechanics.  It  remains  to  us  to  speak  of 
Benedetti's  important  contribution  to  the  doctrine  of  impetus. 

At  the  outset  Benedetti  maintained  that  a  constant  motive  agency 
produced  an  accelerated  motion.  "  In  natural  and  rectilinear  motion 
the  impressio,  the  impetuositas  recepta,  increases  continually,  for  the 
moving  body  contains  in  itself  the  motive  cause,  that  is  to  say  the  pro 
pensity  to  take  itself  to  the  place  to  which  it  is  assigned.  Aristotle 
should  not  have  said  that  a  body  moves  more  rapidly  as  it  approaches 
its  goal,  but  rather  that  a  body  moves  more  rapidly  as  it  becomes  further 
separated  from  its  point  of  departure.  For  the  impressio  increases  pro 
portionally  as  the  natural  motion  is  prolonged,  the  body  continually 
receiving  a  new  impetus.  Indeed  it  contains  in  itself  the  cause  of  motion, 
which  is  the  tendency  to  regain  the  natural  place  from  which  it  has  been 
torn  by  violence.  "  This  qxiotation  shows  that  even  if  Benedetti  remain 
ed  impregnated  with  Aristotle's  ideas,  he  was  not  imprisoned  by  them. 
As  we  shall  see,  he  was  also  able  to  amend  Buridan's  thesis. 

Benedetti  believed  that  the  entity  which  was  conserved  in  motion 
was  the  impetus  in  a  straight  line.  In  his  opinion  a  horizontal  wheel, 
as  exactly  symmetrical  as  possible  and  resting  on  a  single  point,  cannot 
have  a  perpetual  motion  of  rotation.  He  gives  four  different  reasons 
for  this. 

The  first  is  "  that  such  a  motion  is  not  natural  for  the  wheel.  " 

The  second  is  because  of  the  friction  at  the  support. 

The  third,  because  of  the  resistance  of  the  air. 

The  fourth  reason,  which  is  the  only  truly  important  one,  we  shall 
quote  from  Benedetti's  text.1 

66  We  consider  each  of  the  corporeal  parts  which  moves  on  its  own 
by  means  of  the  impetus  which  has  been  imparted  to  it.  This  part  has 
a  natural  tendency  to  rectilinear  motion,  not  to  a  curvilinear  one.  If  a 

1  Translated  into  French  by  DUHEM. 


XVIth   CENTURY  105 

particle  chosen  on  the  circumference  of  the  aforesaid  wheel  was  cut  off 
from  this  body,  there  is  no  doubt  that,  at  a  certain  time,  this  detached 
part  would  move  in  a  straight  line  through  the  air.  We  can  see  this  in 
the  example  of  the  slings  which  are  used  to  throw  stones.  In  these 
slings  the  impetus  of  motion  which  has  been  imparted  to  the  projectile 
describes,  by  a  kind  of  natural  propensity,  a  rectilinear  path.  The  stone 
which  is  thrown  sets  out  on  a  rectilinear  path  along  the  line  which  is 
tangent  to  the  circle  which  it  describes  at  the  outset,  and  which  touches 
this  circle  at  the  point  at  which  the  stone  was  released,  as  it  is  reasonable 
to  assume.  " 

In  short,  Benedetti  was  the  first  to  have  clarified  the  idea  that 
the  impetus  was  conserved  in  a  straight  line.  From  this  correct  idea, 
however,  he  formed  an  incorrect  conclusion.  Thus  he  maintained 
that  the  motion  of  a  wheel  must  slow  down  spontaneously,  because  its 
particles  do  not  follow  the  rectilinear  paths  which  they  have  an  innate 
tendency  to  take. 

In  fact,  when  the  Schoolmen  of  the  XlVth  Century  applied  their 
doctrine  of  impetus  indiscriminately  to  rectilinear  and  curvilinear 
motions,  they  confused  two  notions  which  a  classical  science  should 
have  distinguished  ;  the  principle  of  inertia,  or  the  conservation,  in 
certain  privileged  connections,  of  the  rectilinear  uniform  motion  of 
an  isolated  material  point ;  and  the  principle  of  energy,  which  entails 
the  conservation  of  the  living  force  when  these  forces  do  no  work. 
If  he  had  the  essential  merit  of  having  caught  a  glimpse  of  the  principle 
of  inertia,  Benedetti,  on  the  other  hand,  misunderstood  part  of  the 
truth  of  Buridan's  thesis. 

8.  GIORDANO  BRUNO  (1548-1600)  AND  THE  COMPOSITION  OF  MOTION. 

Giordano  Bruno  is  best  known  as  a  metaphysician.  A  remote 
disciple  of  Nicholas  of  Cues,  he  believed  at  the  same  time  in  the  unity 
and  the  infinity  of  worlds.  He  illustrated  this  by  means  of  a  system 
of  Monads  which  were  at  once  material  and  spiritual,  which  were  not 
born  and  did  not  perish,  but  combined  with  and  separated  from  each 
other.  He  was  burnt  alive  at  Rome  on  February  17th,  1600  for  his 
lampooning  of  the  Papacy  rather  than,  it  seems,  for  his  metaphysical 
ideas. 

Bruno,  who  taught  at  the  College  of  France  and  accepted  Coper 
nicus'  system,  was  a  determined  adversary  of  aristotelian  ideas.  Thus 
he  rebutted,  in  his  Cena  de  le  Ceneri  (1584),  Aristotle's  objection  to 
the  motion  of  the  Earth  which  had  depended  on  the  fact  that  a  stone 
thrown  vertically  upwards  fell  again  at  its  starting-point.  This  he 


106  THE    ORIGINS 

accomplished   by   an   argument  which  was   analogous   to;   but   more 
precise  than,  that  of  Oresme. 

For  this  purpose,  he  visualises  two  men,  one  on  the  deck  of  a  ship 
and  the  other  on  the  bank,  and  each  holding  a  stone  in  his  hand.  It 
is  arranged  that,  at  some  instant,  the  hands  are  in  sensibly  the  same 
position  and  that,  then,  the  stones  are  allowed  to  fall  simultaneously 
on  the  deck  of  the  ship.  The  second  man's  stone  will  fall  behind  that 
of  the  first.  For  the  stone  belonging  to  the  man  on  the  ship  "  moves 
with  the  same  motion  as  the  ship.  It  has  therefore  a  certain  virtus 
impressa  which  the  other  does  not  possess  .  .  .  even  though  the  stones 
have  the  same  gravity ;  though  they  traverse  the  same  air  ;  though 
they  start  from  points  which  are,  as  nearly  as  can  be  arranged,  the 
same  ;  though  they  are  subject  to  the  same  initial  impact.  " 

9.  BERNARDINO  BALDI  (1553-1617).  STATICS  AND  GRAVITY 
"  EX  VIOLENTIA.  " 

Bernardino  Baldi  was  at  once  a  theologian,  archeologist,  linguist  and 
geographer.  He  was  a  familiar  of  Guido  Ubaldo  and,  in  mechanics, 
seems  to  have  been  influenced  by  Leonardo  da  Vinci  and  others.  In 
1582  he  wrote  Exercitationes  in  mechanica  Aristotelis  problemata  which 
was  not  printed  until  1621.  Bernardino  Baldi  rejected  the  point  of 
view  of  virtual  velocities — that  of  Aristotle — in  statics.  u  We  cannot 
be  sure  that  the  admirable  effect  of  a  lever  has  as  its  cause  the  velocity 
which  follows  from  the  lengths  of  the  arms.  Indeed,  what  is  the 
velocity  of  something  that  does  not  move  ?  Now  the  lever  and  the 
balance  do  not  move  when  they  are  in  equilibrium  and  nevertheless 
a  small  power  can  then  support  a  large  weight.  It  will  be  retorted 
that  if  a  very  great  velocity  is  not  apparent  in  very  long  arms,  it  will 
at  least  be  potentially  present.  Now  the  force  which  maintains  [the 
lever]  maintains  the  action.  " 

In  a  more  positive  way  Baldi  concerned  himself  with  the  equili 
brium  of  a  tripod  and,  in  this  connection,  gave  the  rule  of  the  polygon 
of  sustentation.  He  took  the  product  of  the  weight  of  a  body  and 
the  height  of  the  centre  of  gravity  as  a  measure  of  the  effort  necessary 
to  overturn  the  body.  He  also  discovered  the  correct  law  for  the 
stability  of  the  balance  and  made  a  study  of  the  sensitivity  of  the 
balance.  He  accepted  Leonardo's  solution  of  the  problem  of  the  in 
clined  plane  without  rectifying  it. 

In  dynamics  Baldi  distinguished  between  gravity  by  nature  and 
gravity  by  violence,  in  which  the  influence  of  an  external  motive 
agency  was  concerned.  In  a  projectile  animated  by  a  simple  motion 


XVIth    CENTURY  107 

of  translation,  the  centre  of  the  natural  gravity ,  J3,  coincides  with  the 
centre  of  the  gravity  ex  violently  under  the  influence  of  an  impulsion 
with  direction  BD.  These  two  centres  are  "  only  distinct  by  rea 
soning  and  not  in  reality.  "  And  Baldi  adds — 

"  Projectiles  cease  to  move  because  the  im 
pression  whose  nature  and  impetuosity  governs 
them  is  in  no  way  natural,  but  purely  acci 
dental  and  violent.  Now  nothing  which  is 
violent  is  perpetual.  ...  As  long  as  violence 
predominates,  violent  motion  is  entirely  similar 
to  natural  motion — it  is  lower  at  the  start ; 
later,  by  the  very  fact  of  the  motion,  it  becomes 
more  rapid  ;  then,  as  the  impressed  violence 
weakens  bit  by  bit,  it  slows  down ;  finally 

the  motion  disappears  at  the  same  time  as  the  impetus  and  the  moving 
body  comes  to  rest.  " 

As  Duhem  has  remarked,1  "  this  opinion  is  strange  and  not  very 
logical.  If  one  can  assume  that  the  natural  gravity,  which  is  a  per 
manent  motive  agency,  creates  at  each  instant  a  new  impetus,  one 
cannot  conclude  from  this  that  the  artificial  gravity,  that  is  the  impetus 
imparted  by  the  motive  agency,  engenders  in  its  turn  an  impetus  of  a 
second  kind.  "  However  strange  it  may  be  this  thesis,  handed  on 
by  Mersenne,  was  to  be  taken  over  by  Roberval.  Duhem  has  even 
followed  its  trail  as  far  as  Descartes.  "  2 

1  Studes  sur  Leonard  de  Fmci,  Vol.  I,  p.  139. 

2  Cf.  a  letter  from  DESCARTES  to  MERSENNE  on  April  26th,  1643,  which  discusses 
the  question  of  whether  a  sword  thrust  is  more  effective  if  it  is  made  with  the  point, 
the  central  part  or  that  near  the  hilt  of  the  sword. 


CHAPTER    SEVEN 


XVIth  CENTURY 

(Continued) 
XVIIth  CENTURY 

TYCHO-BRAHE  AND  KEPLER 


1.  THE   SYSTEM   DUE  TO   TYCHO-BRAHE  (1546-1601). 

While  the  students  of  mechanics  of  the  Renaissance  remained 
faithful  to  the  Schoolmens'  tradition  and  rehearsed  their  arguments 
without  taking  account  of  the  observations  that  were  available  to 
them  the  astronomers  were  patiently  accumulating  a  host  of  data 
that  were  to  be  seized  by  classical  science  for  the  formulation  of  the 
laws  of  dynamics.  Tycho-Brahe  occupies  a  prominent  place  among 
these  observers  because  of  the  volume  and  the  precision  of  his  obser 
vations,  which  were  the  foundation  upon  which  Kepler's  laws  were 
based.  We  must  say  a  word  here  of  his  system  of  the  World  and  of 
his  ideas  on  dynamics. 

Tycho-Brahe  rejected  Ptolemy's  system  because  of  the  complexity 
of  its  epicycles.  He  rejected  Copernicus'  system  on  the  grounds  that 
the  comets  observed  in  opposition  to  the  Sun  were  not  affected  by  the 
annual  motion  of  the  Earth. 

In  his  Astronomiae  instauratae  progymnasmata  (1582)  he  wrote, 
"  That  heavy  mass  of  the  earth,  so  ill-disposed  towards  motion,  cannot 
be  displaced  and  agitated  in  this  way  without  conflicting  with  the 
principles  of  physics.  The  authority  of  the  Holy  Scriptures  opposes 
it.  ...  I  have  set  out  to  examine  seriously  whether  there  is  any 
hypothesis  which  is  completely  in  accord  with  the  phenomena  and  the 
mathematical  principles  without  being  repugnant  to  physics  and  without 
incurring  the  censures  of  theology.  It  has  turned  out  as  I  had  hoped.  .  . . 

"  I  believe,  firmly  and  without  reservation,  that  the  motionless 
earth  must  be  placed  at  the  centre  of  the  World,  in  accord  with  the 


XVIth   AND    XVIIth    CENTURIES  109 

feelings  of  ancient  astronomers  or  physicists  and  the  testimony  of  the 
Scriptures.  I  in  no  way  assume,  like  Ptolemy  and  the  Ancients,  that 
the  earth  is  the  centre  of  the  orbits  of  the  secondary  moving  bodies. 
Rather  I  believe  that  the  celestial  motions  are  arranged  in  such  a  way 
that  only  the  Moon  and  the  Sun  and  the  Eighth  Sphere — the  most 
distant  of  all — have  the  centres  of  their  motions  at  the  earth.  The  five 
other  planets  turn  round  the  Sun  as  round  their  Chief  and  King,  and 
the  Sun  is  always  at  the  centres  of  their  spheres  and  is  accompanied  by 
them  in  its  annual  motion.  Thus  the  Sun  will  be  the  law  and  the 
end  of  all  these  revolutions  and,  like  Apollo  among  the  Muses,  it 
alone  will  determine  all  the  celestial  harmony  of  the  motions  which 
surround  it.  " 

Tycho-Brahe's  initial  faith  in  his  system  is  embodied  in  the  following 
formula.  "  Nova  mundani  systematis  hypotyposis  ab  authors  nuper 
adinventa  qua  turn  vetus  ilia  Ptolemaica  redundantia  et  inconcinnitas, 
turn  etiam  recens  Coperniana  in  motu  terrae  physica  absurditas  excluduntur, 
omniaque  apparentiis  caelestibus  aptissime  correspondent.  " 

However,  this  assurance  is  less  obvious  in  a  letter  written  to  Roth- 
mann  and  dated  February  21st,  1589,  "  If  you  prefer  to  make  the 
earth  and  the  seas,  together  with  the  moon,  revolve ;  if  you  wish  that 
the  earth,  however  ill-suited  to  motion  and  far  below  the  stars  it  may 
be,  behave  like  a  star  in  the  ethereal  regions,  you  are  certainly 
the  master.  .  .  .  But  are  not  earthly  things  being  confused  with 
celestial  things  ?  is  not  the  whole  order  of  nature  being  turned  upside- 
down?  " 

Fundamentally  it  was  religious  prejudice  that  dictated  the  form  of 
Tycho-Brahe's  thesis,  for  he  was  too  wideawake  not  to  admit  the  super 
iority  of  the  copernican  system  over  that  of  Ptolemy.  "  I  acknowledge 
that  the  revolution  of  the  five  planets,  which  the  Ancients  attributed 
to  epicycles,  are  easily  and  at  little  cost  explained  by  the  simple  motion 
of  the  Earth  ;  that  the  mathematicians  have  adopted  many  absurdities 
and  contradictions  which  Copernicus  set  aside  ;  and  that  his  system 
even  agrees  a  little  more  accurately  with  celestial  phenomena.  " 

In  order  that  the  planets  might  turn  about  the  Sun,  Tycho-Brahe 
was  obliged  to  assume  that  the  rotation  of  the  Sun  round  the  Earth 
was  due  to  an  attraction  that  was  different  from  that  between  the  planets 
and  the  Sun. 

In  dynamics  he  opposed  the  motion  of  the  Earth  with  the  objection 
that  a  stone  dropped  from  the  top  of  a  tower  fell  at  the  bottom.  Thus 
he  did  not  appreciate  the  fallacies  in  this  argument,  which  Oresme  and 
Giordano  Bruno  had  already  indicated,  though  he  was  almost  certainly 
unaware  of  their  writings. 


110  THE  ORIGINS 

2.  KEPLER  (1571-1631).     THE  GENERAL  CHARACTER  OF  HIS   CONTRI 
BUTION. 

It  may  seem  strange  that  this  review  of  the  origin  of  mechanics 
should  finish  with  Kepler's  work.  But  if  he  is  numbered  among  the 
classics  for  his  three  fundamental  laws  on  the  motion  of  the  planets, 
his  metaphysical  tendencies  and  his  ideas  on  dynamics  place  him  in 
the  scholastic  tradition.  Though  a  forerunner  of  Newton,  his  own 
inspiration  were  the  writings  of  Nicholas  of  Cues. 

Kepler's  character  is  most  complex.  A  tireless  calculator,  he 
returned  to  the  interpretation  of  observations  without  ever  being  discou 
raged,  and  rejected  every  law  that  allowed  the  slightest  imprecision. 
With  great  wisdom  he  remarked  that  in  the  domain  of  Astronomy 
innovations  were  apt  to  lead  to  absurdities.  By  this  he  meant  that  the 
observations  of  the  Ancients,  however  rough,  should  not  be  neglected. 
Though  a  disciple  of  Tycho-Brahe,  he  had  no  less  respect  for  Ptolemy 
and  was  alive  to  the  necessity  of  not  adhering  to  the  copernican  system. 
His  preconceived  ideas,  his  errors,  inconsistencies  and  illusions  are  not 
hidden  from  the  reader.  Occasionally  his  writings  have  the  air  of  the 
confessional —  thus  he  declares  that  his  desire  to  succeed  makes  him 
blind,  "  cum  essem  caecus  pro  cupiditate. " l  He  compares  scientific 
truth  to  a  nymph  who  steals  away  after  allowing  herself  to  be  seen,  and 
quotes  Virgil  2  in  this  connection.  We  see  him  sacrificing  himself  to 
metaphysics,  seeking  the  reflection  of  preordained  harmonies  on  every 
occasion,  and  even  lending  himself  to  astrology.  Should  we  regard  this 
as  a  fashion  of  the  time,  or  as  evidence  of  difficulties  of  quite  another 
kind  ?  Indeed,  the  following  declaration  is  attributed  to  Kepler — 
Astronomy  would  die  of  hunger  if  her  daughter,  Astrology,  did  not 
earn  enough  bread  for  two.  .  .  . 

Kepler's  first  scientific  work,  Mysterium  Cosmographicum,  was 
published  at  Tubingen  in  1596.  The  pythagorean  influence  which  was 
to  become  apparent  in  all  Kepler's  thought  emerges  clearly  from  this 
youthful  work.  Thus  he  sought  to  incorporate  the  dimensions  of  the 
different  planetary  orbits  into  the  copernican  system  by  comparing 
them  with  the  radii  of  spheres  inscribed  or  circumscribed  to  five  regular 
polyhedra.  He  assumed  that  the  planets  moved  under  the  influence 
of  an  anima  matrix  localised  in  the  Sun,  whose  action  on  the  planets 

1  Astronomia  nova,  p.  215. 

2  "  Malo  me  Galataea  petit,  lasciva  puella 

Et  fugit  ad  salices,  et  se  cupit  ante  videri.  " 

The  question  here  is  the  discovery  of  the  elliptic  trajectories  of  the  planets,  Astro 
nomia  nova,  p.  283. 


XVIth   AND   XTIIth   CENTURIES  111 

was  greater  as  they  came  nearer  to  the  Sun.  This  property,  confined 
to  the  plane  of  the  ecliptic,  is  therefore  inversely  proportional  to  the 
distance.  The  same  is  true  of  the  velocity  produced,  in  accordance 
with  the  aristotelian  dynamics  to  which  Kepler  remained  faithful. 

At  least  Mysterium  Cosmographicum  had  the  merit  of  attracting  the 
interest  of  Tycho-Brahe,  who  thereupon  used  Kepler  in  the  analysis  of 
planetary  observations  and,  if  the  tradition  is  to  be  believed,  charged 
Kepler  with  the  task  of  preparing  a  new  table  of  the  planets.  Kepler 
completed  this  task  in  1627,  with  the  publication  of  Tabulae  Rudolphinae. 


3.  THE  ORIGIN  OF  THE  LAW  OF  AREAS. 

We  shall  now  follow  Kepler's  fundamental  work  in  theoretical  astro 
nomy —  Astronomia  nova  ocmoAoy^ro  g,1  seu  Physica  Caelestis  tradita 
commentariis  de  motibus  stellae  Martis  ex  observationibus  G.  V.  Tychonis 
Brake,  Prague,  1609. 

In  this  work  Kepler  seeks  a  theory  of  Mars  which  will  take  account 
of  the  observations  in  a  precise  way  and  which  will  be,  at  the  same  time, 
compatible  with  the  systems  of  Ptolemy,  Copernicus  and  Tycho-Brahe, 

The  following  is  a  very  much  abbreviated  version  of  Kepler's  demon 
stration  of  the  law  of  areas  in  the  case  of  eccentrics.2 

In  the  figure  a  is  the  centre  of  the  World^  that  is,  the  Sun  in  the  coper- 
nican  system  and  the  Earth  in  the  other  astronomical  systems. 

The  centre  of  the  eccentric  which  the  planet  describes  is  at  /?.  (This 
term  refers  to  the  Earth  in  the  copernican  system  and  the  Sun  in  the 
others.) 

The  point  y  is  the  equant  (punctum  aequantis),  or  the  point  about 
which,  according  to  Ptolemy's  hypothesis,  the  "  planet  "  appears  to 
describe  a  circle  with  uniform  velocity.  Kepler  draws  this  circle  as  a 
dotted  line  with  the  point  y  as  centre  and  radius  equal  to  that  of  the 
eccentric  of  centre  /?. 

Further,  like  Ptolemy,  Kepler  assumes  the  bisection  of  the  eccentricity, 
or  that  a/3  is  equal  to  fiy. 

Starting  from  the  aphelion  (or  apogee)  8  and  the  perihelion  (or 
perigee)  £,  two  very  small  arcs  dip  and  sco  are  drawn  in  such  a  way  that 
the  points  y>,  a  and  o>  are  colinear.  Then  the  lines  yyj  and  yo>  are  drawn 
to  cut  the  dotted  circle  in  %  and  r  respectively. 

"  According  to  Ptolemy,  since  the  entire  circle  vcp  (equal  to  the  eccen- 

1  That  is,  "  concerning  the  search  for  causes,  "  meaning  at  once  dynamical  and 
metaphysical  causes. 

2  Astronomia  nova,  Chapter  XXXII,  p.  165. 


112 


THE   ORIGINS 


trie  but  with  centre  y  (is  a  measure  of  the  planet's  period,  then  the 
arc  v%  will  be  a  measure  of  the  time  the  planet  (mora)  spends  on  the 
arc  &p  of  the  eccentric.  "     Kepler  calls  the  arc  5y  arcus  itineris  and  the 
arc  v%9  arcus  temporis*    The  same  is  true  for  the  arcs  ea)  and  cpr* 
Having  supposed  the  angle  (5oc^  to  be  very  small,  Kepler  writes 

7^       the  arc  VY        ,    ys       the  arc  SGI 
(1)  .  —J         — 


the  arc 


the  arc  rep 


Fig.  34 


XVIth  AND    XVIIth   CENTURIES  113 

Because  of  the  bisection  of  the  eccentricity,  the  length  fid  is  the 
arithmetic  mean  of  yd  and  a<5.  But  the  arithmetic  mean  of  two  quan 
tities  which  are  nearly  equal  to  each  other  is  just  greater  1  than  their 
geometric  mean — this  Kepler  verifies  by  means  of  a  numerical  example. 
Then 

/9<5  (or  yv)  <x.d 

=^=  and   >  —  • 

yd  fid 

From  which  follows  through  eq  (1) 

^-^ 

,  ^  the  arc  vy  ,         <x.d 

(2)  4  =£=  and  >  — • 

the  arc  ipS  fid 

In  the  same  way,  if  /?£  is  the  arithmetic  mean  between  and  ye  and 
oce,  it  is  found  that 

ys  ^e 

=£=  and   <  — • 


fie  (or  y(p)  oce 

Hence,  by  (1), 

the  arc  sa>  Be 

-  ^  =£±  and   <  —  • 

the  arc  <pr  as 

If  then  one  considers,  on  the  eccentric,  two  very  small  arcs  dip  and 
£O>,  assumed  to  be  equal  to  each  other,  each  of  them  will  be  the  mean  pro 
portional  between  the  arc  v%  —  the  time  spent  at  the  aphelion  —  and  the 

<•—  «v 

arc  (pr  —  the  time  spent  at  the  perihelion.     Further,  the  ratio  of  the 

^  ^      .  //?e\2 

arc  v%  to  the  arc  (pr  will  be  very  nearly  equal  to  (  —  1  • 

Or  again,  more  clearly,  if  two  very  small  and  equal  arcs  dy  and  £O> 

are  taken  on  the  eccentric,  the  ratio  of  the  times  spent  on  the  arcs  will 

t> 

be  the  ratio  of  the  arcs  v%  and  9?r,  and  will  be  equal  to  —  ,  since,  to  the 

.       ,  .        a<5        fe0\* 

square  in  the  eccentrics,    —  =  —  )  . 
^ 


Now  Kepler  is  in  a  position  to  state  the  law  of  areas  for  eccentrics. 

"  Quanto  longior  est  oc(5  quam  oce,  tanto  diutius  moratur  Planeta  in 
certo  aliquo  arcui  excentrici  apud  6,  quam  in  aequali  arcu  excentrici  apud  s." 

That  is,  the  greater  a<5  is  than  as,  the  longer  the  planet  will  remain 
on  a  certain  arc  in  the  immediate  neighbourhood  of  d  than  on  an  equal 
arc  of  the  eccentric  in  the  neighbourhood  of  £. 

1  In  the  modern  sense,  "  approximately  equal  to  and  greater  than  "  or  "  =s=  and  >.'* 


114  THE    ORIGINS 

In  the  neighbourhoods  of  other  points  on  the  eccentric  which  are 
opposite  to  each  other  with  respect  to  the  centre  of  the  world  a,  the 
behaviour  of  the  planet  is  analogous,  "  quanto  evidentior  in  demonstra- 
tione?  tanto  minor  in  effectu.  " 

In  fact,  Kepler  confined  himself  to  the  remark  that  the  proportion 
of  oc/^  to  OLV  is  smaller,  and  that  of  a0  to  oa  is  much  smaller,  than  the 
proportion  of  oc<5  to  oca.  (See  fig.  34.) 

Kepler  translated  this  purely  geometrical  and  kinematic  analysis 
into  dynamical  terms  in  the  very  title  of  the  chapter  which  we  have 
analysed —  Virtutem  quam  Planetam  movet  in  circulum  attenuari  cum 
discessu  afonte.  (The  strength — understood  as  the  force — by  means  of 
which  the  Planet  moves  circularly  falls  off  with  the  distance  from  the 
source  [of  motion].)  This  is  evidence  of  the  fact  that  Kepler  remained 
faithful  to  aristotelian  dynamics.  Indeed,  the  force  is  measured  in 
Kepler's  mind  by  the  inverse  of  a  duration  of  sojourn  on  an  arc,  that 
is,  by  the  velocity  to  which  it  corresponds.  We  see  here  the  continuity 
of  Kepler's  views  from  his  Mysterium  Cosmographicum  to  his  Astro- 
nomia  nova. 


4.  ORIGIN  OF  THE  LAW  OF  THE  ELLIPTICITY  OF  PLANETARY  TRAJECTORIES. 

Tycho-Brahe  and  Longomontanus  had  prepared  a  table  of  the  oppo 
sitions  of  Mars  since  1580.  Tycho-Brahe,  who  had  started  Kepler  on 
his  study  of  the  theory  of  Mars,  himself  represented  the  orbit  of  that 
planet  by  an  eccentric  whose  geometrical  centre  did  not  bisect  the  eccen 
tricity. 

Now  Kepler,  either  by  tradition  or  because  of  his  metaphysics,  was 
attached  to  the  hypothesis  of  the  bisection  of  the  eccentricity,  which 
Ptolemy  had  put  forward  in  connection  with  the  major  planets  alone. 
He  even  went  as  far  as  to  extend  it  to  the  Earth's  orbit  (in  the  context 
of  Copernicus'  system)  and  to  that  of  the  Sun  (in  the  other  systems). 

Kepler  immediately  started  a  methodical  refinement  of  the  values 
assigned  to  the  radii  of  the  Earth's  orbit,  which  determined  the  scale 
of  all  the  other  interplanetary  distances. 

He  then  turned  his  attention  to  Mars.  Being  unable  to  follow  him 
through  all  the  various  detours  that  he  made,  we  shall  only  record  that 
he  succeeded  in  accounting  for  all  of  twelve  oppositions  of  Mars  to 
within  2'  of  arc.  This  was  accomplished  by  a  painful  method  of  trial 
and  error  in  which  four  longitudes  of  Mars  in  opposition  were  used 
simultaneously.  This  necessitated,  on  Kepler's  own  confession,1  no 

1  Astronomia  nova,  Chapter  XVI,  p.  95. 


XVIth   AND   XVIIth   CENTURIES  115 

less  than  seventy  repetitions  of  the  calculation.  But  the  longitudes  of 
Mars  in  position  other  than  opposition  invalidated  the  eccentric  calcu 
lated  in  this  way.  Moreover,  the  eccentric  did  not  satisfy  the  hypothesis 
of  the  bisection  of  the  eccentricity — it  transpired  that  the  distances 
from  the  geometrical  centre  to  the  Sun  and  to  the  equant  were  not 

7-232 

equal,  but  were  in  the  ratio  — 

^  11-332 

Returning  to  the  hypothesis  of  the  bisection  of  the  eccentricity  for 
the  orbit  of  Mars,  and  relying  on  the  observation  of  the  opposition  of 
Mars  in  1613,  Kepler  found  an  error  of  about  8'  in  the  annual  parallax 
of  the  planet.  Fortunately  for  theoretical  astronomy  and  for  the  deve 
lopment  of  newtonian  mechanics,  Kepler 
refused  to  neglect  such  a  disparity 
between  calculation  and  observation. 
He  proceeded  to  evaluate  the  distances 
from  Mars  to  the  Sun  in  terms  of  the 
distances  from  the  Earth  to  the  Sun. 
The  accompanying  diagram  shows  how 
a  knowledge  of  the  longitudes  of  Mars 
and  of  the  Earth,  together  with  a 
knowledge  of  the  two  radii  (SB  and  SC) 
of  the  Earth's  orbit,  allow  the  distance 
from  the  Sun  to  Mars  (SM)  to  be  de 
termined.  In  the  diagram  the  circle 
with  centre  0  represents  the  Earth's  Fi£*  35 

orbit,  S  the  Sun  and  M,  Mars,  while  B 

and  C  are  two  positions  of  the  Earth  for  the  same  position  of  Mars. 
In  particular,  Kepler  proceeded  in  this  way  for  the  distances  from  Mars 
to  the  Sun  in  the  neighbourhood  of  the  aphelion  and  of  the  perihelion, 
and  thus  obtained  the  eccentricity  of  the  planet. 

Kepler  compared  these  observations  with  a  circular  eccentric  satis 
fying  the  principle  of  the  bisection  of  the  eccentricity.  He  established  a 
systematic  failure  of  distances  with  respect  to  the  circumference  of  the 
circle,  "  Itaque  plane  hoc  est ;  orbita  planetae  non  est  circulus,  $ed  ingre- 
diens  at  latera  utraque  paulatim,  iterumque  ad  circuit  amplitudinem  in 
perigeo  exiens,  cujusmodi  figuram  itineris  ovalem  appellitant.  "  x 

Only  observation  could  make  Kepler  give  up  the  hypothesis  of  the 
circle,  which  was  based  on  the  authority  of  the  ancients  and,  for  the 
rest,  agreed  with  his  own  metaphysics. 

At  first  Kepler  was  reluctant  to  make  an  ellipse  of  this  oval  orbit, 

1  Astronomia  nova.  Chapter  XLIV,  p.  213. 


116  THE    ORIGINS 

though  he  did  investigate  whether  a  particular  ellipse  that  he  had 
chosen  could  reconcile  the  data,  only  to  discover  that  this  was  not  so.  1 
Finally,  however,  after  many  unsuccessful  attempts,  he  wrote,  "  Inter 
circulum  vero  et  ellipsin,  nihil  mediat  nisi  ellipsis  alia  "  (between  a  circle 
and  an  ellipse  there  can  be  nothing  but  a  second  ellipse).  And  he  con 
cludes  that  "  Ergo  ellipsis  est  Planet  ae  iter.  "  2 

5.  KEPLER'S  THIRD  LAW. 

The  extremely  important  positive  success  of  the  theory  of  Mars  did 
not  turn  Kepler's  interest  away  from  astrology  and  metaphysics.  Thus 
more  than  ten  years  elapsed  before  Harmonices  Mundi  was  published 
at  Linz  in  1619.  Of  the  first  five  books  in  this  work  only  the  last  makes 
mention  of  astronomy,  and  even  this  is  confused  with  strange  meta 
physical  conceptions.  For  example,  we  see  him  develop  an  analogy 
between  the  angular  velocities  of  the  planets  about  the  Sun  and  the 
frequencies  of  musical  notes,  and  expressing  the  oscillation  of  these 
angular  velocities  during  the  course  of  a  revolution  by  means  of  a  musical 
notation.  The  question  is  really  one  of  pythagorean  harmony,  with 
the  reservation  that  it  remains  abstract  and  that  Kepler  did  not  pretend 
that  it  was  perceptible  by  our  senses. 

In  following  the  Astronomia  nova  we  have  seen  that  Kepler  carne 
across  the  law  of  areas  before  that  of  ellipticity.  It  has  however  become 
customary  to  reverse  the  order  in  which  these  two  laws  are  presented, 
and  to  forget  that  Kepler,  without  justification,  extended  the  law  of 
areas  to  elliptical  trajectories  allthough  he  had  only  established  it  for 
eccentrics. 

Kepler's  third  law  is  stated  in  Chapter  III  of  Book  V  of  Harmonices 
MundL3  He  recalls  the  fruitless  efforts  that  he  had  made,  since  the 
beginning  of  his  scientific  career,  to  establish  a  connection  between  the 
periods  of  the  planets  and  the  dimensions  of  their  orbits.  Not  until 
March  8th,  1618,  did  he  come  across  the  characteristic  ratio  in  this  law 
— a  gross  error  of  calculation  made  him  reject  it  at  first.  Finally  he 
persuaded  himself  of  its  correctness — "  Res  est  certissima  exactissimaque, 
quod  proportio  quae  est  inter  binorum  quorumcunque  Planetarum  tempora 
periodica,  sit  praecise  sesquialtera  proportionibus  mediarum  distantiarum^ 
id  est  Orbium  ipsorum.  "  (One  thing  is  absolutely  certain  and  correct, 

3 

that  the  ratio  between  the  periods  of  any  two  planets  is,  to  the  power  ~, 

£ 

1  Astronomia  nova,  Chapter  XLV. 

2  Ibid.,  Chapter  LV,  p.  285. 
8  P.  189. 


XVIth   AND   XVIIth   CENTURIES  117 

exactly  that  of  their  mean  distances,  that  is,  of  their  orbits.)     This 
quite  empirical  result  may  be  written 


6.  KEPLER  AND  THE  CONCEPT  OF  INERTIA. 

Kepler  had  the  merit  of  having  emphasised  the  concept  of  inertia 
more  completely  than  his  predecessors  had  done — indeed  it  is  sometimes 
maintained  that  he  actually  formulated  the  principle  of  inertia.  This 
is  not  true  in  the  sense  that  Kepler's  concept  of  inertia  remained  linked 
with  Aristotle's  mechanics  and  with  Buridan's  doctrine  as  modified  by 
the  German  School  of  the  XVth  Century. 

"  The  proper  characteristic  of  material  which  forms  the  greatest 
part  of  the  Earth  is  the  inertia.  Motion  is  repugnant  to  it,  and  more  so 
as  a  great  quantity  of  material  is  confined  in  a  smaller  volume.  "  1 

Kepler  adds — 

"  This  material  inertia  of  a  terrestrial  body,  this  density  of  the  same 
body,  constitute  exactly  the  subject  on  which  the  impetus  of  rotational 
motion  is  impressed.  It  is  impressed  there  exactly  as  in  a  top  which 
turns  because  of  violence.  The  heavier  the  material  of  the  top  is,  the 
better  it  assimilates  the  motion  impressed  by  the  external  force  and  the 
more  lasting  this  motion  is.  "  2 

Kepler's  dynamics  follows  directly  from  the  ideas  of  Nicholas  of  Cues, 
whom  he  called  "  divinus  mihi  Cusanus.  "  He  took  up  the  example  of 
the  toy  top  which  Nicholas  of  Cues  had  given,  and  applied  the  doctrine 
of  impetus  impressus  to  celestial  bodies. 

"  Could  not  God  have  produced  [such  an  impetus  impressus]  in  the 
Earth,  as  from  the  exterior,  at  the  beginning  of  time  ?  It  is  this  im 
pression  which  has  produced  all  the  past  rotations  of  the  Earth  and 
which  maintains  them  even  now,  though  their  number  already  exceeds 
two  millions.  Indeed,  this  impression  keeps  all  its  vigour  because  the 
rotation  of  the  Earth  is  not  hindered  by  impact  or  by  any  external 
roughness  ;  or  by  the  ethereal  fluid,  which  is  devoid  of  density.  No 
more  is  it  hindered  by  any  weight,  or  by  any  internal  gravity.  As  for 
the  inertia  of  the  material,  that  is  the  very  subject  which  receives  the 
impetus  and  conserves  it  as  long  as  the  motion  continues.  "  3 

1  Opera  omnia,  Vol.  VI,  p.  174. 

2  Ibid.,  p.  175. 

3  Ibid.,  p.  176. 


118  THE    ORIGINS 

Kepler  believed  that  the  material  of  the  Earth  was  separated  into 
circular  fibres  whose  centres  were  aligned  with  the  axis  of  rotation. 
"  This  arrangement  of  the  Earth  into  circular  fibres  predisposes  it  to 
the  motion  that  it  receives.  All  the  same,  it  appears  that  these  fibres 
are  the  instruments  of  the  motive  cause  rather  than  the  motive  cause 
itself.  "  i 

The  impetus  communicated  to  the  Earth  by  the  Creator  becomes  a 
soul.  "  It  is  a  soul  of  a  strange  kind.  It  confers  on  the  Earth  neither 
growth  nor  discursive  reason  (sic) —  it  merely  moves  it.  But,  better 
than  a  simple  corporeal  faculty,  this  motive  soul  assures  the  perfect 
regularity  of  diurnal  motion.  This  motion,  indeed,  is  no  longer  a  vio 
lent  motion,  in  any  sense,  for  the  Earth.  What  is  there,  indeed,  more 
natural  to  a  material  than  its  form,  to  a  body  than  its  faculty  or  soul  ?  "  2 

7.  KEPLER  AND  THE  DOCTRINE  OF  ATTRACTION. 

Following  the  example  of  Copernicus,3  Kepler  showed  himself  to  be 
a  Pythagorean  in  the  matter  of  gravitation.  He  therefore  denied  the 
thesis  that  Albert  of  Saxony  had  made  classical  since  the  XlVth  Century. 

"  The  doctrine  of  gravitation  is  erroneous.  A  single  mathematical 
point,  whether  it  be  the  centre  of  the  World  or  any  other  point,  cannot 
effectively  move  heavy  bodies,  nor  be  the  object  towards  which  they 
tend.  Therefore  let  them,  the  Physicists,  prove  that  such  a  force  can 
belong  to  a  point,  which  is  not  a  body  and  which  is  only  conceived  in  an 
entirely  relative  way. 

"  It  is  impossible  that  the  [substantial]  force  of  a  stone,  which  sets 
the  body  in  motion  of  itself,  should  seek  a  mathematical  point,  the 
centre  of  the  World  without  regard  to  the  body  in  which  that  point 
may  be  situated.  Therefore  let  them,  the  Physicists,  establish  that 
natural  things  have  sympathy  for  that  which  does  not  exist.  "  4 

And  Kepler  expounds  "  the  true  doctrine  of  gravity.  " 

"  Gravity  is  a  mutual  affection  between  parent  bodies  (Gravitas  est 
affectio  corporea,  mutua  inter  cognata  corpora)  which  tends  to  unite  them 
and  join  them  together.  The  magnetic  faculty  is  a  property  of  the 
same  kind.  It  is  the  Earth  which  attracts  the  stone,  even  though  it 
might  not  tend  towards  the  Earth.  In  the  same  way,  if  we  place  the 
centre  of  the  Earth  at  the  centre  of  the  World,  it  is  not  towards  the 
centre  of  the  World  that  bodies  are  carried,  but  rather  towards  the 

1  Opera  omnia,  Vol.  VI,  p.  178. 

2  Ibid.,  p.  179. 

8  See  above,  p.  85. 

4  Astronomia  nova,  Introductio,  para.  VIII. 


XVIth  AND    XVIIth   CENTURIES  119 

centre  of  the  body  around  which  they  belong,  that  is  to  say,  the  Earth. 
Also,  the  heavy  bodies  will  be  carried  towards  whatever  place  the  Earth 
is  carried  to,  because  of  the  faculty  which  animates  it. 

"  If  the  Earth  was  not  round,  heavy  bodies  would  not  move  directly 
towards  the  centre  from  all  directions.  But  according  to  whether  they 
come  from  one  place  or  another,  they  will  be  carried  to  different  points. 

"  If,  in  a  certain  position  in  the  World,  two  stones  are  placed  near  each 
other  and  outside  the  sphere  of  attraction  of  all  other  bodies  which 
could  attract  them,  these  stones,  like  two  magnets,  will  tend  to  unite 
in  an  intermediate  position  and  the  distances  they  will  travel  in  order 
to  unite  will  be  in  inverse  ratio  to  their  masses.  "  1 

1  Ibid. 


PART  TWO 


THE  FORMATION  OF  CLASSICAL  MECHANICS 


CHAPTER   ONE 


STEVIN'S  STATICS 
SOLOMON  OF  CAUX 


1.  THE  STATICS  OF  STEVIN  (1548-1620). 

Stevin's  first  work  on  statics  was  published  in  Flemish  at  Leyden 
in  1586,  under  the  title  De  Beghinselen  der  Weegconst*  A  more  complete 
version  appeared  in  1605.  Finally,  in  1608,  Stevin  united  these  works 
under  the  title  of  Hypomnemata  Mathematica.  This  work  was  trans 
lated  into  French  as  early  as  1634. 

Stevin's  statics  is  developed  geometrically  in  a  manner  similar  to 
that  used  by  Archimedes. 

In  it,  the  author  systematically  neglects  "  the  motions  of  machines, 
formed  of  wood  or  iron,  in  which  certain  parts  are  lubricated  with  oil 
or  lard,  others  are  swollen  by  the  humidity  of  the  air  or  corroded  with 
rust,  in  which  these  varied  circumstances  and  also  many  others  some 
times  facilitate  the  motion,  sometimes  hinder  it.  " 

Moreover,  Stevin  refuses  to  consider  the  excess  of  motive  power 
which  motion  demands,  "  for  the  obstacles  to  motion  have  no  certain 
and  unique  relation  with  the  object  moved.  " 

Still  more  rigorously,  Stevin  rejected  the  consideration  of  arcs 
of  a  circle  described  by  the  ends  of  the  arms  in  the  problem  of  the 
equilibrium  of  a  lever.  And  he  justified  this  by  means  of  a  syllogism. 
"  Something  which  does  not  move  does  not  describe  a  circle.  Two 
weights  in  equilibrium  do  not  move.  Therefore  two  weights  in  equi 
librium  do  not  describe  circles.  " 

We  see  that  Stevin  eschewed  the  point  of  view  of  virtual  velocities 
in  order  to  romp  in  the  field  of  pure  statics.  At  least  he  imposed  this 
restriction  on  the  form  of  his  writing.  He  was  not,  however,  to  main 
tain  it  exclusively,  as  we  shall  show. 

On  the  subject  of  the  lever,  Stevin  added  some  further  refinements 
to  Archimedes'  demonstrations  which  we  shall  pass  over. 


124  THE    FORMATION    OF   CLASSICAL   MECHANICS 

He  solved  the  problem  of  the  equilibrium  of  a  heavy  body  on  an 
inclined  plane  by  a  method  that  was  completely  original  and  which 
was  based  on  the  impossibility  of  perpetual  motion. 

This  is  his  demonstration,  taken  from  the  French  edition  of  1634. 

"  Given.  —  Let  ABC  be  a  triangle  whose  plane  is  perpendicular 
to  the  horizon  and  whose  base  AC  is  parallel  to  the  horizon.  Let 
a  weight  D  be  placed  on  the  side  AB,  which  is  to  be  twice  J3C,  and 
a  weight  JS,  equal  to  D,  be  placed  on  the  side  BC. 


"  The  Requirement.  —  It  is  necessary  to  show  that  the  power 
(or  capacity  of  exerting  power)  of  the  weight  E  is  to  that  of  the  weight 
D  as  AB  is  to  BC,  that  is,  as  2  is  to  1. 

"  Construction.  —  Round  the  triangle  let  there  be  arranged  a 
system  of  fourteen  spheres  equal  in  weight,  size,  and  equidistant  from 

each  other  at  the  points  D,  E,  F P,  Q,  B,  and  threaded  on  a  cord 

passing  through  their  centres  in  such  a  way  that  there  are  two  spheres 
on  BC  and  four  on  BA ....  Let  S,  T,  V be  three  fixed  points  on  which 
the  cord  can  run  freely  without  being  caught. 

"  Demonstration.  —  If  the  power  of  the  weights  D,  JR,  Q,  P  were  not 
equal  to  the  power  of  the  weights  E,  F,  one  of  the  sides  would  be  heavier 
than  the  other.  Suppose  then  that  the  four  D,  jR,  Q,  P  are  heavier 
than  the  two  E9  L.  Now  the  four  0,  N,  M,  F  are  equal  to  the  four 
G,  H,  /,  K.  Now  the  side  with  eight  spheres  D,  R,  Q,  P,  0,  JV,  M,  L 


STEVIN   AND    CAUX 


125 


will  be  heavier  than  that  of  the  six  spheres  JS,  .F,  G,  H,  I,  K  and,  since 
the  heavier  part  will  dominate  the  lighter,  the  eight  spheres  will  fall 
and  the  six  will  rise.  Thus  D  will  come  to  where  0  is  at  present  and 
the  others  will  do  the  same.  That  is,  that  £,  F,  G,  H  come  to  the 
positions  where  P,  Q,  .R,  D  are  now  and  JT,  K  to  where  jE,  jF  are.  How 
ever  the  effect  of  the  spheres  will  have  the  same  disposition  as  pre 
viously  and  for  the  same  reason  the  eight  spheres  will  weigh  more 
and,  when  they  fall,  will  make  eight  others  come  in  their  place.  Thus 
this  motion  will  have  no  end,  which  is  absurd.  The  demonstration  will 
be  the  same  in  the  opposite  case.  Therefore  the  part  jD,  R  .  .  .  .  L 
of  the  ring  will  be  in  equilibrium  with  the  part  J5,  JF,  ....  1C.  If  there 
be  taken  away  from  both  sides  the  heavinesses  which  are  equal  and 
similarily  arranged,  the  four  spheres  0,  JV,  M,  L  on  the  one  hand  and 
the  four  G,  fl,  I,  K  on  the  other,  the  four  D,  It,  (),  P  which  will  be  left 
will  be  in  equilibrium  with  the  two  .E,  F.  Hence  E  will  have  a  power 
twice  that  of  D.  Therefore  the  power  of  E  is  to  that  of  D  as  the  side 
BA9  let  it  be  2,  is  to  the  side  BC,  let  it  be  1.  " 

Stevin  had  read  Cardan  and  referred  to  his  Opus  novum.  He  could 
not  have  been  ignorant  of  De  Subtilitate  in  which  Cardan,  after  Leonardo 
da  Vinci,  held  that  perpetual  motion  was  impossible. 

Stevin  was  legitimately  proud  of  his  demonstration.  He  reproduced 
the  associated  diagram  in  the  frontispiece  of  Hypomnemata  Mathema- 
tica  with  the  legend  "  Wonder  en  is  gheen  wonder  "  (The  magic  is  not 


Fig.  37 


magical),  no  doubt  intending  to  indicate  that  he  had  logically  explained 
a  fact  instead  of  invoking  magic  as  the  Greeks  had  done,  before  Archi 
medes,  in  connection  with  levers. 

From  this  theorem  on  the  inclined  plane  Stevin  deduced  the  value 
of  the  weight  E  which  could  support  the  column  D  on  an  inclined 


126 


THE    FORMATION    OF    CLASSICAL   MECHANICS 


plane  by  means  of  a  thread  parallel  to  the  plane  and  starting  from 
the  centre  of  gravity  of  the  column.     The  result  was  given  by 


D 

~E 


AB 
BC 


He  then  studied  a  series  of  more  complicated  examples,  like  the 
following  one  in  which  the  direct  elevation,  M,  in  equilibrium  with 
the  column  D  is  compared  with  an  oblique  elevation,  such  as  E,  which 
is  able  to  hold  the  column  on  the  inclined  plane. 

In  these  circumstances  M  is  equal  to  D,  and  the  preceding  result 

v    w      D        AB 

is  applicable  :  -=  =  ^7=  • 


E 


CB 
BC 


DI 


AT 

N°W  ^ 


m      *      M      DL 

Therefore- =  — 


Fig.  38 


Thus  Stevin,  after  many  false  starts,  arrived  at  an  enunciation 
and  even  a  verification  of  the  rule  of  the  parallelogram  of  forces  for 
the  particular  instance  in  which  the  two  forces  are  at  right  angles. 

Let  a  column  of  centre  of  gravity  C  be  hung  from  the  points  D 
and  E  by  means  of  two  strings  CD,  CE.  Complete  the  parallelogram 
CHIK  whose  diagonal  CI  is  vertical. 

"  The  direct  elevation  is  to  the  oblique  elevation  as  CI  is  to  CH.  But 
the  direct  elevation  CI  is  equal  to  the  weight  of  the  column.  Therefore 
the  weight  of  the  whole  column  to  the  weight  which  occurs  at  D  is  as 
CI  is  to  CjfiT.  In  the  same  way,  the  weight  which  occurs  at  E  will 
be  found  by  producing  the  line  IK  from  I  parallel  to  DC  to  meet  CJE. 


STEVIN   AND    CAUX 


127 


In  words,  the  weight  of  the  column  which  occurs  at  E  will  be  as  the 
straight  elevation   CI  is  to  the  oblique  elevation  CK.  " 


Fig.  39 

From  these  considerations  Stevin  deduced  the  tension  of  the  threads 
of  a  funicular  polygon  and  thus  became  the  originator  of  graphical 
statics. 


2.  STEVIN  AND  THE  PRINCIPLE  OF  VIRTUAL  WORK. 

Returning  to  the  point  of  view  which  had  turned  statics  into  a 
purely  deductive  science,  and  starting  only  from  the  assumption  of 
the  impossibility  of  perpetual  motion,  Stevin  quite  clearly  stated  the 
principle  of  virtual  work.  This  occurs  in  volume  IV  of  his  Hypomnemata 
in  connection  with  Stevin's  work  on  the  equilibrium  of  systems  of 
pulleys. 

"  The  distance  travelled  by  the  force  acting  is  to  the  distance  tra 
velled  by  the  resistance  as  the  power  of  the  resistance  is  to  that  of 
the  force  acting.  (Ut  spatium  agentis  ad  spatium  patientis,  sit  potentia 
patientis  ad  potentiam  agentis) .  " 


128  THE  FORMATION  OF  CLASSICAL  MECHANICS 

3.  STEVIN'S  HYDROSTATICS. 

Stevin's  contribution  to  hydrostatics  is  quite  remarkable.  He 
clearly  stated  the  principle  of  solidification  according  to  which  a  solid 
body  of  any  shape  and  of  the  same  density  as  a  given  fluid  can  remain 
in  it  at  equilibrium  whatever  its  position  may  be,  and  without  the 
pressures  in  the  rest  of  the  fluid  being  modified.  He  used  this  principle 
to  determine  the  pressure  on  each  element  of  the  base  by  solidifying 
all  the  liquid  except  that  in  a  narrow  channel  abutting  on  this  element, 
and  verified  that  this  pressure  was  independent  of  the  shape  of  the 
receptacle  and  depended  only  on  the  weight  of  the  column  of  liquid 
which  filled  the  channel.  This  led  him  to  state  the  hydrostatic  paradox — 
that  a  fluid,  by  means  of  its  pressure,  can  exert  a  total  effort  on  the 
bottom  of  a  vessel  which  can  be  considerably  greater  than  the  total 
weight  of  the  fluid.  He  also  determined  the  resultant  of  the  pressures 
on  an  inclined  plane  boundary  wall  by  dividing  this  surface  into  hori 
zontal  slices  and  passing  to  the  limit  by  increasing  the  number  of 
slices  indefinitely. 

Finally,  he  related  Archimedes'  principle  to  the  impossibility 
of  perpetual  motion.  Thus  he  was  guided  by  the  same  idea  as  in 
the  problem  of  the  inclined  plane. 

4.  SOLOMON  OF  CAUX  (1576-1630)  AND  THE  CONCEPT  OF  WORK. 

Solomon  of  Caux  was  a  practical  Norman  who  was  concerned 
with  the  construction  of  hydraulic  screws.  In  1615,  after  having 
read  Cardan,  he  published  at  Frankfurt  a  work  entitled  The  reasons 
of  moving  forces  together  with  various  machines,  as  much  useful  as  plea 
sant ,  to  which  are  added  some  designs  of  grottoes  and  fountains. 

It  is  to  this  author  that  we  owe  the  term  work  in  the  sense  that  it 
is  used  in  mechanics  now. 


CHAPTER   TWO 


GALILEO  AND  TORRICELLI 


1.  GALILEO'S  STATICS. 

Galileo  (1564-1642)  started  his  scientific  career  in  the  way  that 
was  customary  in  his  time,  by  annotating  Aristotle's  De  Caelo.  His 
manuscript  remained  unpublished  until  1888  and  is  a  typical  scholastic 
document,  even  though  it  refers  to  certain  moderns  like  Cardan  and 
Scaliger. 

Nevertheless,  holding  a  chair  of  mathematics  at  the  University 
of  Pisa  at  the  early  age  of  twenty- five,  Galileo  was  not  long  in  causing 
a  scandal  by  publicly  experimenting  on  the  fall  of  heavy  bodies,  by 
attacking  his  elders,  and  by  offending  a  natural  protector  like  John 
de  Medici  by  wounding  his  pride  in  his  inventions.  However,  his 
fierce  intellectual  independence,  which,  in  its  turn,  was  to  earn  him 
many  rebuffs,  developed  very  rapidly.  He  did  not  long  remain  a 
slave  to  the  scholastic  discipline. 

We  shall  not  concern  ourselves  here  with  Galileo's  biography, 
which  belongs  to  the  general  history  of  science,  but  shall  make  an 
analysis  of  his  contribution  to  statics  and  dynamics. 

First  we  shall  follow  the  Mechanics  of  Galileo  in  the  French  version 
which  Mersenne  published  at  Paris  in  1634.  Chronologically  this 
work  lies  between  the  manuscript  of  Galileo's  lectures  at  Padua  in 
1594  and  the  treatise  Delia  Scienza  meccanica  which  was  printed  at 
Ravenna  in  1649,  seven  years  after  its  author's  death. 

At  the  beginning  of  the  Mechanics  Galileo  emphasises  "  that  ma 
chines  are  useful  for  manoeuvring  great  loads  without  dividing  them, 
because  often  there  is  much  time  and  little  force  .  .  .  but  he  who  would 
shorten  the  time  and  use  only  a  little  force  will  deceive  himself.  " 
That  is,  Galileo  considers  the  product  of  the  force  and  the  velocity 
in  conformity  with  Aristotle's  thesis. 

To  Galileo  the  heaviness  of  a  body  was  a  "  natural  inclination 
of  the  body  to  take  itself  to  the  centre  of  the  Earth.  " 


D 


130  THE    FORMATION   OF    CLASSICAL   MECHANICS 

The  moment  was  the  inclination  of  the  same  body  considered  in 
the  situation  which  it  occupied  on  the  arm  of  a  lever  or  a  balance. 
It  is  "  made  up  of  the  absolute  heaviness  of  the  body  and  its  separation 
from  the  centre  of  the  balance,  and  corresponds  to  the  Greek  QOTZIJ.  " 
The  existence  of  a  centre  of  heaviness  (centre  of  gravity)  of  a  body 
was  just  as  much  an  experimental  fact  to  Galileo  as  it  was  to  the  School 
men.  "  Each  body  principally  weighs  through  the  centre  in  which 
it  masses  and  unites  all  its  impetuosity  and  weight.  " 

In  turn,  Galileo  studied  the  lever,  the  steelyard,  the  lathe,  the  fly 
wheel,  the  crane,  the  winch,  the  pulley  and  the  screw. 

The    discussion    of   the    screw    entailed    a    study   of  the  inclined 

plane  and,  in  this  connection,   Galileo 
/•     was  able  to  do  better  than   his  prede 
cessors  had  done. 

He  envisages  a  perfectly  round  and 
polished  ball  to  be  placed  on  a  per 
fectly  smooth  surface.  On  the  hori 
zontal  plane  AB  "  the  ball  is  indifferent 
to  motion  and  rest,  so  that  the  wind  or 

the  smallest  force  can  move  it.  "     But 
w  — — — ^  u 

a  greater  lorce  is  necessary  in  order  to 

Flg>  40  lift  the  ball  on  the  inclined  planes  AC, 

AD,  AE  and  finally,  "  it  will  only  be 

possible  to  lift  the  ball  on  the  perpendicular  plane  with  a  force  equal 
to  its  whole  weight.  " 

Galileo  proceeds  by  considering  two  equal  weight,  A  and  C,  in 
equilibrium  on  the  lever  ABC.  In  this  way  he  is  able  to  amend  Pappus's 
demonstration,  which  he  cites  in  this  connection.  If  the  arm  BC 
falls  to  JBF,  the  moment  of  the  weight  F  becomes  less  than  the  moment 

BK 
of  the  equal  weight  A  in  the  ratio   p~^* 


"  When  the  weight  is  at  F  it  is  partly  maintained  by  the  circular 
plane  CI  and  its  slope — or  the  tendency  which  it  has  to  the  centre 
of  the  earth  is  diminished  by  the  extent  that  BC  exceeds  BK.  So 
that  it  is  supported  by  this  plane  to  the  same  extent  as  if  it  had  been 
supported  by  the  tangent  GFH?  more  especially  as  the  slope  of  the 
circumference  at  the  point  F  only  differs  from  the  slope  of  the  tangent 
GFH  by  the  insensible  angle  of  contact.  " 

By  means  of  this  remarkable  artifice  Galileo  reduces  the  effect 
of  the  weight  F  on  the  inclined  plane  GFH  to  the  effect  of  the  same 
weight  suspended  as  if  from  the  arm  of  the  lever  BF.  And  he  concludes 
that  "  the  ratio  of  the  total  and  absolute  moment  of  the  moving  body, 


GALILEO   AND   TORRICELLI 


131 


in  the  perpendicular  to  the  horizon,  to  the  moment  which  it  has  on 
the  inclined  plane  HF  is  the  same  as  the  ratio  of  FH  to  FK. 

BF  .  ,FH 

For         1S  6q  ~FK.' 


Galileo  also  solves  the  problem  of  the  inclined  plane  by  appealing, 
this  time,  to  the  concept  of  virtual  work. 

"imagine  that  in  the  triangle  ABC  the  line  AB  represents  the 

horizontaf  plane,  the  line  AC  the  inclined  plane  whose  height  mil 

be  measured  by  CB.     On  the  plane  AC 

is   placed  a  body   E   attached  to   the 

string  EDF.    At  F  the  string  carries  a 

weight,  or  a  force,  which  is  related  to 

the  weight  E  in  the  ratio  of  the  line 

BC  to  the  line  CA.    If  the  weight  F 

starts  to  fall,  drawing  the  body  E  along 

the   inclined  plane,  the   body    E   will 

travel  a  path  in  the  direction  of  AC 

which  is  equal  to  that  which  the  heavy 

body  F  describes  in  its  fall.     But  the 


Fig.  42 


body          escres  n  . 

following  observations  are  necessary.     It  is  true  that  the  body  E 
have  travelled  all  the  line  AC  in  the  same  time  that  the  weight  F 
have  SL  to  fall  an  equal  distance.     But  during  this  time,  the  body 
E  will  not  have  been  separated  from  the  common  centre  of  heavy 


132 


THE    FORMATION    OF    CLASSICAL   MECHANICS 


things  by  a  distance  greater  than  the  vertical  J5C,  while  the  weight  F, 
falling  vertically,  will  have  fallen  a  distance  equal  to  the  whole  line  AC. 
Now  the  bodies  only  resist  an  oblique  motion  to  the  extent  to  which 
they  are  taken  away  from  the  centre  of  the  earth We  can  legitima 
tely  say  that  the  path  of  the  force  F  keeps  the  same  ratio  to  the  path 
of  the  force  E  as  the  ratio  of  the  length  AC  to  the  length  CJ5,  and  is 
therefore  equal  to  the  ratio  of  the  weight  E  to  the  weight  F.  " 

2.  GALILEO  AND  THE  FALL  OF  BODIES. 

Thanks  to  a  letter  which  Galileo  wrote  to  Paolo  Sarpi,  dated  October 
16th,  1604,  x  we  know  that  as  early  as  this  Galileo  believed  in  the  now 
classical  law  of  distances  5  =  constant  X  t2. 

"  The  distances  gone  through  in  natural  motion  are  in  square 
ratio  to  the  times  of  fall.  Consequently  the  distances  travelled  in 
equal  times  are  related  to  each  other  like  the  consecutive  odd  numbers 
starting  from  unity.  " 

Nevertheless,  at  first  Galileo  associated  this  law  of  distances  with 
an  incorrect  law  of  velocities,  namely  v  =  k  -  5. 
"We  recall  that  as  early  as  the  XlVth  Century 
Albert  of  Saxony  had  hesitated  between  this 
law  and  the  correct  one,  v=k*t. 

To  Galileo,  the  law  v  =  k  •  s  was  explained 
in  the  following  way. 

"  A  body  which  moves  naturally  increases 
in  velocity  to  the  extent  that  it  is  separated 
from  the  source  of  its  motion.  " 

The  arguments  by  which  Galileo  sought  to 
verify  these  two  laws  simultaneously  are 

\k  "  rather  odd.     Certainly  they  are  incorrect,  but 

they  show  us  the  development  of  his  thought 
and  deserve  to  be  quoted  as  showing  what 
detours  he  made  before  he  became  eman 
cipated  from  them. 

"  If  the  heavy  body  starts  from  the  point  A 
and  falls  along  the  line  AB,  I  suppose  that 
the  degree  of  velocity  at  the  point  D  exceeds 
the  degree  of  velocity  at  the  point  C  in  the 

ratio  of  DA  to  CA  ;  that  in  the  same  way,  the  degree  of  velocity  at  E 
is  to  the  degree  of  velocity  at  D  as  EA  is  to  DA.  Thus,  at  every 
point  of  AB  the  body  will  have  a  velocity  proportional  to  the 

1  The  Works  of  Galileo,  Italian  National  Edition,  Vol.  X,  p.  115. 


f 


B 


7 


43 


GALILEO   AND   TORRICELLI  133 

distance  from  this  same  point  to  the  origin  A.  This  principle  appears 
to  me  to  be  very  natural.  It  corresponds  to  all  the  observations  that 
we  make  of  machines  whose  purpose  is  hitting.  Given  this  principle, 
I  will  demonstrate  the  rest. 

"  Let  the  line  AK  make  any  angle  with  the  line  AF  and,  through 
the  points  C,  D,  E,  F  let  the  parallels  CG,  DH,  EI9  FK  be  drawn. 
Since  the  lines  FK,  El,  DH,  CG  have  the  same  relation  to  each  other  as 
the  lines  FA,  EA,  DA,  CA,  the  velocities  at  the  points  F,  E,  D,  C  are 
therefore  related  to  each  other  like  the  lines  FK,  El,  DH,  CG.  There 
fore  the  degrees  of  velocity  at  all  the  points  of  the  line  AF  are  constantly 
increasing  according  to  the  increasing  of  the  parallels  drawn  from 
these  same  points. 

"  Moreover,  since  the  velocity  with  the  moving  body  goes  from 
A  to  D  is  made  up  of  all  the  degrees  of  velocity  acquired  at  all  the 
points  of  the  line  AD,  and  since  the  velocity  with  which  it  has  travelled 
the  line  AC  is  made  up  of  all  the  degrees  of  velocity  acquired  at  all 
the  points  of  the  line  AC,  the  ratio  of  the  velocity  with  which  it  has 
travelled  the  line  AD  to  the  velocity  with  which  it  has  travelled  the 
line  AC  is  that  between  all  the  parallels  drawn  from  all  the  points 
of  the  line  AD  to  the  line  AH.  This  [ratio]  is  that  of  the  triangle 
ADH  to  the  triangle  ACG,  that  is,  the  ratio  of  the  square  of  AD  to  the 
square  of  AC.  Therefore  the  relation  of  the  velocity  with  which  the 
moving  body  has  travelled  the  line  AD  to  the  velocity  with  which  it 
has  run  through  the  line  AC  is  the  square  of  the  ratio  of  DA  to  CA. 

"  But  the  ratio  of  a  velocity  to  a  velocity  is  the  inverse  of  the  ratio 
of  the  corresponding  times,  for  the  time  decreases  when  the  velocity 
increases.  The  ratio  of  the  duration  of  motion  along  AD  to  the  dur 
ation  of  motion  along  AC  is  therefore  the  square  root  of  the  ratio  of  the 
distance  AD  to  the  distance  AC.  Therefore  the  distances  from  the 
starting  point  are  as  the  squares  of  the  times.  However,  the  distances 
travelled  in  equal  times  are  to  each  other  as  the  consecutive  odd  numbers 
starting  from  unity.  This  is  in  accord  with  what  I  have  always  said 
and  with  observations  made.  All  the  verities  are  thus  in  accord.  "  l 

Briefly,  from  the  inexact  hypothesis  that  v  =  k  •  s,  Galileo  obtains 

^       l     .      v(D)        DA 
the  relation  -77^  =  -— • 
v(C)        CA 

Then  by  a  consideration  of  a  series  of  parallels  erected  from  each 

f      ATX    1.        J     J  •  1  T.  ,        -  VAD  DA* 

point  ol  AM  he  deduces,  incorrectly,  the  relation 


v(AC)         \CA 
where  v(AD)  and  v(AC)  are  the  mean  velocities  on  AD  and  AC. 

1  Complete  Works  of  Galileo,  Italian  Edition,  Florence  1908,  Vol.  VIII,  p.  373. 


134  THE    FORMATION   OF   CLASSICAL   MECHANICS 

From  the  last  relation,  and  again  incorrectly,  he  concludes  that 
the  ratio 


t(AC}~  V  CA 
and  thus  arrives  at  the  correct  law 

5  =  constant  X  t2. 

We  shall  now  follow  the  treatise  Discorsi  e  dimostrazioni  matema- 
tiche  intorno  a  due  nuove  scienze  attenanti  alia  Meccanica  ed  i  movimenti 
localL  The  first  edition  of  this  work  appeared  in  1638  and  was  later 
supplemented  by  the  author,  although  these  additions  only  appear 
in  an  edition  that  was  printed  at  Bologna  in  1655. 

In  these  Discorsi  three  characters,  Salviati  (Galileo),  Sagredo  (a 
Venetian  senator  and  friend  of  Galileo)  and  Simplicio  (who  represents 
Scholasticism)  discuss  the  work.  This  dialogue  form  has  the  obvious 
inconvenience  of  making  the  book  difficult  to  read,  but  has  the  in 
estimable  advantage  of  allowing  the  author  to  show  the  development 
of  his  thought. 

The  Text  declares  that  the  fall  of  bodies  is  uniformly  accelerated, 
or  that  the  "  increase  of  the  velocity  is  like  that  of  the  time.  "  Starting 
from  rest,  the  moving  body  receives  equal  degrees  of  velocity.  This 
the  Text  assumes  a  priori. 

"  Why  indeed  not  believe  that  the  increases  in  velocity  follow 
the  most  simple  and  banal  law  ?  " 

Refusing  to  lose  himself  in  the  discussions  which  had  occupied 
the  Schoolmen,  Salviati  brushes  aside  all  argument  on  the  cause  of 
the  fall  of  bodies. 

He  recalls  that  he  had,  for  some  time,  believed  that  the  velocity 
could  increase  as  the  distance  did.  As  we  have  just  seen,  this  was 
his  opinion  in  1604.  He  now  rejects  this  belief. 

"  If  the  velocities  are  proportional  to  the  distances  travelled,  the 
distances  will  be  travelled  in  equal  times.  Therefore,  if  the  velocities 
with  which  the  body  travelled  the  4  cubits  were  double  those  with 
which  it  travelled  the  first  two  cubits  (as  the  distances  are  doubled)  the 
durations  of  travel  will  be  equal.  But  the  same  moving  body  can  only 
travel  the  4  or  the  2  cubits  in  the  same  time  if  this  motion  is  instan 
taneous.  Now  it  is  apparent  that  the  motion  of  a  heavy  body  lasts  a 
certain  time,  and  that  it  travels  the  first  two  cubits  in  less  time  than 
the  four.  Therefore  it  is  not  true  that  its  velocity  increases  as  the 
distance.  " 


GALILEO   AND    TORRICELLI 


135 


We  note  here,  with  Jouguet,1  that  this  argument  of  Galileo  is  not 
quite  correct.  The  law  v  =  k-s  immediately  leads  to  s  =  s0exp(kt). 
In  order  that  there  should  be  motion  it  is  necessary  that,  contrary  to 
the  hypothesis,  s0  should  be  different  from  zero  when  t  =  0.  Other 
wise  it  is  necessary  to  assume  that  in  the  first  instant  the  body  travels 
the  distance  s0  instantaneously. 

Given  this,  Salviati  makes  the  following  postulate.  "  I  assume  that 
the  degrees  of  velocity  acquired  by  the  same  moving  body  on  differently 
inclined  planes  are  equal  whenever  the  heights  of  the  planes  are  equal.  " 

The  moving  body  is  assumed  to  be  perfectly  smooth  and  the  planes 
to  be  perfectly  polished. 

In  order  to  substantiate  this  principle  Salviati  starts  from  the  follow 
ing  postulate,  in  which  Galileo's  physical  intuition  is  very  apparent. 

"  Imagine  that  this  sheet  of  paper  is  a  vertical  wall,  that  a  nail  is 
fixed  in  it  and  that  a  ball  of  lead  weighing  an.  ounce  or  two  is  hung 


from  the  nail  by  a  thread  AB.  The  thread  is  to  be  two  or  three  cubits 
long,  perpendicular  to  the  horizon  and  at  a  distance  of  about  two 
fingers  from  the  wall.  Draw  a  horizontal  CD  on  the  wall  to  cut  the 
thread  AB  squarely.  Draw  aside  the  thread  AB  and  the  ball  into  the 
position  AC.  Then  release  the  ball.  We  will  see  this  descend,  describ 
ing  the  arc  CB,  and  pass  the  extremity  B  in  such  a  way  that  it  will  go 
up  again,  along  J3D,  almost  to  the  line  CD  which  has  been  drawn. 
Each  time  there  will  be  a  small  deficiency,  and  this  circumstance  is 
precisely  due  to  the  resistance  of  the  air  and  of  the  thread.  From  this 
we  can  conclude,  in  all  truth,  that  the  impeto  at  the  point  B  which  is 

1  JOUGUET,  L.  M.,  Vol.  I,  p.  96. 


136  THE   FORMATION    OF   CLASSICAL   MECHANICS 

acquired  by  the  ball  in  its  descent  of  the  arc  CB  is  such  that  it  suffices 
to  make  it  remount  the  identical  arc  BD  to  the  same  height.  When 
this  observation  has  been  repeated  again  and  again,  fix  in  the  wall  a 
nail  which  projects  about  five  or  six  fingers,  exactly  opposite  the  vertical 
AB — for  example,  at  E  or  at  F.  The  ball  will  describe  the  arc  CB,  the 
thread  turning  as  before.  When  the  ball  comes  to  I?,  the  thread  will 
tangle  in  the  nail  E  and  the  ball  will  be  obliged  to  travel  the  circum 
ference  BG  which  has  E  as  centre.  Then  we  see  that  this  can  produce, 
at  the  extremity  JB,  the  very  impeto  that  can  make  the  moving  body 
rise  again  along  the  arc  BD  until  it  almost  reaches  the  horizontal  CD. 
Now,  gentlemen,  you  will  see  with  pleasure  that  the  ball  attains  the 
horizontal  at  the  point  G.  The  same  thing  would  happen  if  the  nail 
were  fixed  lower,  at  F  for  example.  The  ball  will  describe  the  arc  B I 
and  will  always  finish  its  ascent  on  the  line  CD.  And,  if  the  nail  were 
too  low  for  the  ball  to  attain  the  height  CD  (this  would  happen  if  the 
nail  were  nearer  jB  than  CD)  the  thread  would  wrap  itself  round  the 
nail.  This  observation  prevents  one  from  doubting  the  truth  of  the 
principle  that  has  been  supposed.  Since  the  two  arcs  CJ?,  DB  are  equal 
and  similarly  placed,  the  momenta  acquired  at  jB  along  CB  suffices  to 
make  the  same  body  rise  again  along  BD.  Therefore  the  momenta 
acquired  along  D.B  is  equal  to  that  which  would  make  the  same  moving 
body  rise  again,  along  the  same  arc,  from  JB  to  D.  So  that  in  general, 
the  momenta  acquired  along  any  arc  is  equal  to  that  which  can  make 
the  same  body  rebound  along  the  same  arc.  But  all  the  mamenti  which 
make  the  body  rebound  along  the  arcs  J5D,  B  G,  BI  are  equal,  since  they 
are  produced  from  the  momento  acquired  in  the  descent  CB,  as  obser 
vation  shows.  Therefore  all  the  mamenti  acquired  in  descending  the 
arcs  DB,  GJB,  IB  are  equal.  " 

Salviati  goes  on  to  consider  motions  along  variously  inclined  planes. 

"  We  cannot  show  with  the  same  clarity  that  the  same  thing  will 
happen  when  a  perfect  ball  falls  along  inclined  planes  that  are  drawn 
along  the  chords  of  these  same  arcs.  On  the  contrary,  since  the  planes 
form  an  angle  at  the  point  J59  it  is  plausible  that  the  ball,  having  descend 
ed  along  the  chord  CjB  and  meeting  an  obstacle  at  the  bottom  of  the 
planes  which  mount  along  the  chords  JBD,  BG,  J5J,  will  lose  a  part  of 
its  impeto  in  rebounding,  and  will  not  be  able  to  ascend  again  to  the 
height  of  the  line  CD.  But  since  the  obstacle  raised  in  this  way  prevents 
the  observation,  it  seems  to  me  that  the  mind  will  go  on  believing  that 
the  impeto  (which  contains,  indeed,  the  force  of  the  whole  fall)  will  be 
able  to  make  the  body  go  up  again  to  the  same  height.  Therefore  take 
this  assertion  as  a  postulate  for  the  moment — its  absolute  truth  will 
be  established  later  when  we  shall  see  that  the  conclusions  depending 


GALILEO   AND    TORRICELLI 


137 


Fig.  45 


on  this   hypothesis  are,  in  detail,  in  conformity  with  observation.  " 

Galileo  then  established  the  now  classical  laws  of  falling  bodies.  In 
particular,  we  shall  describe  how,  going  back  on  his  opinion  of  1604,  he 
established  the  law  of  velocities. 

66  Since,  in  a  accelerated  motion,  the  velocity  is  continuously  augment 
ed,  the  degrees  of  the  velocity  cannot  be  divided  into  any  determinate 
number.  For  since  the  velocity  changes  from  mo 
ment  to  moment  and  increases  continuously,  they  are 
of  infinite  number.  However,  we  can  represent  our 
intention  better  by  constructing  a  triangle  ABC, 
taking  as  many  equal  parts  AD,  DE,  EF,  FG  as 
we  please  on  the  side  AC,  and  in  drawing  straight 
lines  parallel  to  the  base  BC  through  the  points  D, 
E,  F,  G.  Then,  if  the  parts  marked  on  the  line  AC 
are  equal  times,  we  assume  that  the  parallels 
drawn  through  the  points  D,  J5,  F  represent  the 
degrees  of  the  accelerated  velocity,  degrees  which 
increase  equally  in  equal  times.  .  .  . 

"  But  because  the  acceleration  is  continuous 
from  moment  to  moment  and  not  of  a  discontinuous 
kind  of  this  or  that  duration  .  .  .  before  the  moving 
body  attains  the  degree  of  velocity  DH  that  is  acquired  in  the  time  AD, 
it  has  passed  through  an  infinity  of  smaller  and  smaller  degrees  gained 
in  the  infinite  number  of  instants  that  the  time  AD  contains  and  which 
correspond  to  the  infinity  of  points  that  lie  on  the  line  DA.  However, 
in  order  to  represent  the  infinity  of  degrees  of  velocity  that  precede  the 
degree  DJEf,  it  is  necessary  to  imagine  an  infinity  of  lines,  always  smaller 
and  smaller,  which  should  be  drawn  from  the  various  of  the  infinite 
number  of  points  of  the  line  DA.  In  ultimo,  this  infinity  of  lines  will 
represent  the  surface  of  the  triangle  AHD. 

64  Complete  the  whole  parallelogram  AMBC  and  produce  as  far  as 
the  side  JBM,  not  only  the  parallels  which  have  been  drawn  in  the 
triangle,  but  also  the  infinite  number  of  parallels  that  was  imagined  to 
start  from  all  the  points  of  the  side  AD.  The  line  BC,  which  is  the 
longest  parallel  drawn  in  the  triangle,  represents  the  highest  degree  of 
the  velocity  acquired  by  the  moving  body  in  its  accelerated  motion. 
The  total  surface  of  the  triangle  is  the  mass  and  the  total  of  all  the  velocity 
with  which  the  body  has  travelled  such  a  distance  in  the  time  AC.  In 
the  same  way  the  parallelogram  will  be  the  mass  and  the  union  of  degrees 
of  velocity  each  of  which  is  equal  to  the  maximum  degree  BC.  This 
latter  mass  of  velocities  will  be  twice  the  mass  of  the  increasing  velocities 
of  the  triangle,  because  the  parallelogram  is  twice  the  triangle.  Conse- 


138  THE    FORMATION   OF    CLASSICAL   MECHANICS 

quently,  if  a  moving  body  takes  degrees  of  an  accelerated  velocity,  in  falling, 
which  conform  to  the  triangle  ABC,  and  if  it  passes  through  such  a  distance 
in  such  a  time,  it  will,  when  moving  uniformly,  travel  twice  the  distance 
that  it  has  travelled  in  the  accelerated  motion.  " 

By  an  analogous  argument  whose  detailed  reproduction  would  serve 
no  useful  purpose,  Galileo  arrived  at  the  following  theorem. 

"  If  a  body  starts  from  rest  and  moves  with  uniformly  accelerated 
motion,  the  time  that  it  takes  to  travel  a  certain  distance  is  equal  to 
the  time  that  the  same  body  would  take  to  travel  the  same  distance 
with  a  uniform  motion  whose  degree  of  velocity  was  half  of  the  greatest 
and  final  degree  of  the  velocity  of  the  uniformly  accelerated  motion.  " 

We  know  that  the  Schoolmen,  thanks  to  the  efforts  of  Oresme, 
Heytesbury  and  Soto,  had  already  obtained  this  fundamental  result. 
But  Galileo  did  not  confine  himself  to  the  a  priori  assertion  that  the  fall 
of  bodies  was  uniformly  accelerated.  He  submitted  the  fall  of  a  body 
on  an  inclined  plane  to  an  experiment  which  was,  for  the  time,  per 
formed  in  a  scrupulous  manner  and  repeated  a  hundred  times.  We 
shall  quote  this  essential  passage  of  the  Discorsi,  noting  its  very  marked 
difference  from  the  tendencies  of  purely  rationalist  Scholasticism. 

"  In  the  thickness  of  a  ruler,  that  is,  of  a  strip  of  wood  about  twelve 
cubits  long,  half  a  cubit  wide  and  three  fingers  thick,  a  channel,  a  little 
wider  than  one  finger,  was  hollowed  out.  It  was  made  quite  straight 
and,  in  order  that  it  should  be  polished  and  quite  smooth,  the  inside 
was  covered  with  a  sheet  of  parchment  as  glazed  as  possible.  A  short 
ball  of  bronze  that  was  very  hard,  quite  round  and  well- polished,  was 
allowed  to  move  down  the  channel.  The  ruler,  made  as  we  have  des 
cribed,  had  one  of  its  ends  lifted  to  some  height — of  about  one  or  two 
cubits — above  the  horizontal  plane.  As  I  have  said,  the  ball  was 
allowed  to  fall  in  the  channel  and  the  duration  of  its  whole  journey 
was  observed  in  the  way  that  I  have  explained.  The  same  trial  was 
repeated  many  times  in  order  to  be  quite  sure  of  the  length  of  this  time. 
In  this  repetition,  no  difference  greater  than  a  tenth  of  a  pulse  was  ever 
found.  When  this  observation  had  been  repeated  and  established  with 
precision,  we  made  the  ball  fall  through  only  a  quarter  of  the  length  of 
the  channel,  and  found  that  the  measured  duration  of  fall  was  always 
equal  to  half  of  the  other.  .  . . 

"  When  this  observation  had  been  repeated  a  hundred  times,  the 
distances  travelled  were  always  found  to  be  in  the  ratio  of  the  squares 
of  the  times,  and  this  was  true  whatever  the  inclination  of  the  plane, 
or  that  of  the  channel  in  which  the  ball  fell,  was  made  to  be.  We  also 
observed  that  the  durations  of  fall  on  differently  inclined  planes  were 
in  the  proportion  assigned  to  them  [by  our  demonstrations]. 


GALILEO    AND   TORRICELLI  139 

u  As  for  the  measurement  of  the  time,  a  large  bucket  filled  with 
water  was  suspended  in  the  air.  A  small  hole  in  its  base  allowed  a  thin 
stream  of  water  to  escape,  and  this  was  caught  in  a  small  receptacle 
throughout  the  duration  of  the  ball's  descent  of  the  channel,  or  of 
portions  of  it.  The  quantities  of  water  caught  in  this  way  were  weighed 
on  a  very  accurate  balance.  The  differences  and  relations  of  these 
weights  gave  the  differences  and  relations  of  the  times  with  such  accur 
acy  that,  as  I  have  said,  these  operations  never  gave  a  noticeable  differ 
ence  when  repeated  many  times.  " 

Galileo  then  introduced  the  notion  of  impeto,  which  he  also  called 
talento  and  momenta  del  discendere. 

For  a  given  body,  this  tendency  to  motion  is  greatest  along  the  ver 
tical  BA.  It  is  less  on  the  planes  AD,  AE,  AF.  Finally,  the  impeto 
is  completely  reduced  to  nothing  on  the  horizontal  CA,  where  the  body 
(as  we  have  seen  in  reading  the  Mechanics)  "  is  indifferent  to  motion 
or  to  rest,  and  does  not  of  itself  show  any  tendency  to  move  in  any 
direction  or  any  resistance  to  being  set  in  motion.  "  Salviati  gives  the 
following  explanation  of  this  fact. 

"  In  the  same  way  that  it  is  impossible  that  a  heavy  body,  or  an 
ensemble  of  heavy  bodies  should,  of  its  own  accord,  move  upwards  and 
thereby  go  further  away  from  the  common  centre  to  which  heavy  things 
tend,  so  it  is  impossible  that  it  should  spontaneously  move  if  its  centre 
of  gravity  does  not  approach  the  common  centre  in  its  motion.  There 
fore  the  impeto  of  the  moving  body  will  be  nothing  on  some  chosen 
horizontal,  or  on  a  surface  which  is  equidistant  from  the  aforesaid  centre 
and  is  without  inclination.  " 

This  is  the  Galilean  form  of  the  principle  of  inertia.  Galileo  arrived 
at  it  by  a  kind  of  limiting  process,  starting  from  the  principle  of  virtual 
work. 

Galileo  then  returned  to  the  demonstration  which  he  had  given  of 
the  law  of  heaviness  on  an  inclined  plane,  and  in  which  he  had  appealed 
to  the  principle  of  virtual  work.  He  completed  it,  however,  in  the 
following  way. 

"  Manifestly,  the  resistance,  or  the  smallest  force  which  suffices  to 
stop  or  prevent  a  heavy  body  in  its  descent,  is  as  great  as  the  impeto 
of  that  body.  In  order  to  measure  this  force,  I  shall  make  use  of  the 
gravity  of  another  body.  Imagine  that  a  body  G  rests  on  the  plane 
FA  and  that  it  is  attached  to  a  thread  which  passes  over  F  and  supports 
a  weight  H.  .  .  .  In  the  triangle  AFC9  the  displacement  of  the  body  G, 
for  example  upwards  from  A  to  .F,  is  made  up  of  the  transverse  and 
horizontal  motion  AC  and  the  vertical  motion  CF.  Now  the  resistance 
to  motion  due  to  the  horizontal  displacement  is  zero. .  . .  Consequently 


140 


THE    FORMATION   OF   CLASSICAL   MECHANICS 


the  resistance  is  solely  due  to  the  fact  that  the  body  must  climb  the 
vertical  CF.  Therefore  the  body  G,  moving  from  A  to  F,  only  resists 
because  of  the  vertical  elevation  CF.  But  the  other  body,  H ,  necessarily 
descends  the  whole  length  FA  in  a  vertical  direction.  .  .  .  We  can  there 
fore  say  that  when  equilibrium  is  established  the  moments  of  the  bodies, 
their  velocities  or  tendencies  to  motion,  that  is,  the  distances  which 
they  would  travel  in  the  same  time,  will  be  in  inverse  ratio  to  their 


Fig.  46 


gravities  in  accordance  with  the  law  which  is  true  in  every  instance  of 
motion  in  mechanics.  It  follows  that  to  prevent  the  fall  of  G,  it  will 
suffice  that  H  should  be  so  much  lighter  with  respect  to  G  as  the  distance 
CF  is  less  than  FA.  ...  And  since  we  have  agreed  that  the  impeto  of  a 
moving  body,  the  energy.,  the  moment  or  the  tendency  to  motion  has  the 
same  size  as  the  force,  or  least  resistance,  which  suffices  to  keep  it  still,  we 
conclude  that  the  body  H  is  sufficient  to  prevent  the  motion  of  the  body 
G. . . .  " 

We  note  that  Galileo's  fundamental  idea  consists  in  measuring  the 
impeto,  or  the  tendency  to  motion,  by  means  of  the  static  force  which 
can  be  opposed  to  it.  This  was  an  essentially  original  procedure  which 
had  escaped  the  notice  of  all  the  Schoolmen.  As  Jouguet1  has  legiti 
mately  remarked,  the  same  word  impeto,  in  Galileo's  work,  sometimes 
meant  the  velocity  acquired  by  a  body  in  a  given  time,  and  sometimes  the 
distances  travelled  on  differently  inclined  planes  in  a  certain  time, 
starting  from  rest. 

By  means  of  the  preceding  considerations  Galileo  verified  that  the 
postulate  according  to  which  the  velocities  of  a  body  which  starts  from 

1  L.  M.,  Vol.  I,  p.  106. 


GALILEO    AND   TORRICELLI  141 

rest  and  falls  along  the  line  of  greatest  slope  on  differently  inclined 
planes  of  equal  height  are  the  same  when  it  arrives  at  a  given  horizontal. 
He  also  showed  that  "  if  the  same  moving  body  falls,  starting  from  rest, 
on  an  inclined  plane  and  along  the  vertical  with  equal  height,  the  dura 
tions  of  fall  have  the  same  relation  as  the  lengths  of  the  inclined  plane 
and  of  the  vertical.  " 

This  demonstration  was  necessary  in  order  to  give  full  weight  to  his 
experimental  verification  of  the  law  of  falling  bodies. 

3.  GALILEO  AND  THE  MOTION  OF  PROJECTILES. 

We  have  seen  that  the  Schoolmen  and  those  interested  in  mechanics 
in  the  XVIth  Century  had  only  been  able  to  treat  the  motion  of  projec 
tiles  very  imperfectly.  Galileo  solved  this  problem  by  means  of  a  very 
remarkable  analysis  in  which,  together  with  the  principle  of  inertia, 
there  appears  the  principle  of  the  composition  of  motions  or  of  the  inde 
pendence  of  the  effects  of  forces. 

"We  shall  quote  from  the  text  of  the  Discorsi. 

"  The  Text.  —  I  imagine  a  moving  body  thrown  on  a  horizontal 
plane  without  any  obstacle.  It  is  said  that  its  motion  on  the  plane  will 
remain  uniform  indefinitely  if  the  plane  extends  to  infinity.  But  if  the 
plane  is  limited,  and  if  it  is  set  up  in  air,  when  the  body,  which  we 
suppose  to  be  under  the  influence  of  gravity,  passes  the  end  of  the  plane 
it  will  add  to  the  first  uniform  and  indestructible  motion,  the  downward 
propensity  which  it  has  because  of  its  gravity.  From  this  will  arise  a 
compound  motion,  composed  of  the  horizontal  motion  and  the  naturally 
accelerated  motion  of  descent.  I  call  this  kind  of  motion,  projection. 

"  Animated  by  the  motion  composed  of  a  uniform  horizontal  motion 
and  a  naturally  accelerated  falling  motion,  the  projectile  describes  a 
parabola. 

"  Let  there  be  a  horizontal  or  a  horizontal  plane,  AB,  which  is 
placed  in  air  and  along  which  a  body  moves  uniformly  from  A  to  B. 
At  B,  where  its  support  is  missing,  the  body,  because  of  its  weight,  is 
forced  by  its  gravity  into  a  natural  downward  motion  along  the  vertical 
BN.  Produce  AB  into  the  line  BE,  which  we  shall  use  to  measure  the 
passage  of  time.  Mark  off  equal  lines  BC,  CD,  DE  on  BE,  and  draw 
parallels  to  BN  through  the  points  C,  D,  E.  On  the  first  of  these  paral 
lels  take  an  arbitrary  length  CI ;  on  the  next  one,  a  length  DF  which 
is  four  times  as  great  ;  on  the  third,  a  length  EH  nine  times  greater  ; 
and  so  on,  the  successive  lengths  increasing  as  the  squares  of  CB,  DB, 
EB.  .  .  .  Imagine  that  the  vertical  descent  along  CI  is  added  to  the 
displacement  of  the  body  as  it  is  carried  from  B  to  C  in  uniform  motion. 


142 


THE    FORMATION    OF    CLASSICAL   MECHANICS 


At  the  time  BC  the  body  will  be  at  I.  At  the  time  BD,  which  is  twice 
BC,  its  vertical  distance  of  fall  will  be  equal  to  4CI.  For  it  has  been 
proved  that  the  distances  are  as  the  squares  of  the  times  in  naturally 
accelerated  motions.  In  the  same  way,  the  distance  EH  that  is  tra 
velled  in  the  time  BE  will  be  nine  times  CI,  so  that  the  distances  Eff, 
DF,  CI  are  related  to  each  other  as  the  squares  of  the  lines  EB,  DB, 
CB.  .  .  .  The  points  I,  F,  H  therefore  lie  on  a  parabola.  " 


D 


Fig.  47 


The  discussion  between  the  three  characters  in  the  dialogue  is  of 
considerable  interest.  Sagredo  remarks  that  the  argument  supposes 
that  the  two  motions  combined  in  this  way  "  neither  alter  each  other, 
nor  confuse  each  other,  nor  mutually  hinder  each  other  in  mixing  up.  " 
He  objects  that,  since  the  axis  of  the  parabola  is  vertical  and  goes 
through  the  centre  of  the  earth,  the  particle  will  be  separated  from  this 
centre.  .  .  .  Simplicio  reproaches  the  text  for,  in  the  first  place,  neglec 
ting  the  convergence  of  the  verticals  and,  in  the  second,  neglecting  the 
resistance  of  the  medium. 

Salviati  replies  that,  to  a  first  approximation,  these  objections  may 
be  dismissed.  He  has  experimented  on  a  ball  of  wood  and  one  of  lead 
which  were  arranged  to  fall  from  a  height  of  200  cubits.  The  wooden 
ball,  which  was  more  sensitive  to  the  resistance  of  the  air,  was  not 
noticeably  retarded.  Salviati  recalls  that  the  projectiles  from  firearms 
have  such  velocities  that  their  trajectories  can  be  modified  by  the  re 
sistance  of  the  air. 


GALILEO  AND  TORRICELLI  143 

4.  GALILEO  AND  HYDROSTATICS. 

Galileo  took  up  the  study  of  hydrostatics  in  a  manuscript  called 
Discorso  intorno  alle  cose  che  stanno  in  su  Vacqua  o  che  in  quella  si  muo- 
vono.  This  was  published  at  Florence  in  1612.  Essentially,  his  hydro 
statics  was  based  on  the  principle  of  virtual  velocities,  which  was  directly 
inspired  by  Aristotle's  mechanics.  In  this  work,  Galileo  called  the 
product  of  the  force  and  the  velocity,  momenta. 

"  I  borrow  two  principles  from  the  Science  of  mechanics.  The  first 
is  this —  two  absolutely  equal  weights  that  are  moved  with  equal  velo 
cities  are  of  the  same  power,  or  the  same  momenta,  in  all  their  doings. 

"  To  students  of  mechanics,  momento  means  that  property,  that 
action,  that  efficient  power  by  which  the  motive  agency  moves  and  the 
body  resists.  This  property  does  not  only  depend  on  the  simple 
gravity,  but  also  on  the  velocity  of  motion,  the  different  inclinations 
and  the  different  distances  travelled.  Indeed,  a  heavy  body  produces 
a  greater  impeto  when  it  descends  on  a  very  steep  surface  than  when 
it  descends  on  a  surface  which  is  less  steep.  Whatever  may  be  the 
ultimate  cause  of  this  property,  it  always  keeps  the  name  momento. 

"  The  second  principle  is  that  the  power  of  the  gravitation  increases 
with  the  velocity  of  the  thing  that  is  moved,  so  that  absolutely  equal 
weights  that  are  animated  with  unequal  velocities  have  unequal  powers, 
strengths,  unequal  momenti.  The  more  rapid  is  the  more  powerful,  and 
this  in  the  ratio  of  its  own  velocity  to  the  velocity  of  the  other  weight. . . . 

"  Such  a  compensation  between  the  gravity  and  the  velocity  is 
found  in  all  machines.  Aristotle  has  taken  it  as  a  principle  in  his  Pro 
blems  of  Mechanics.  Hence  the  assertion,  that  two  weights  of  unequal 
size  are  in  equilibrium  with  each  other,  and  have  equal  momenti,  when 
ever  their  gravities  are  in  inverse  ratio  to  the  velocities  of  their  motion, 
may  be  taken  as  wellestablished. " 

In  discussing  the  siphon,  Galileo  remarked  that  a  small  mass  of 
water  contained  in  a  narrow  vessel  could  maintain  in  equilibrium  a 
large  mass  of  water  contained  in  a  wide  vessel,  because  a  small  lowering 
of  the  second  entailed  a  great  increase  in  the  height  of  the  first.  In  this 
respect  Galileo  preceded  Pascal.  If  Duhem  is  to  be  believed,  Galileo 
was  guided  by  a  tradition  that  went  back  to  Leonardo  da  Vinci.1 

The  Discorsi  were  attacked  by  L.  della  Colombe  and  V.  di  Grazia, 
and  defended  by  Benedetto  Castelli  (1577-1644),  a  faithful  disciple  of 
Galileo.  The  same  Castelli  was  the  author  of  a  treatise  on  the  measu 
rement  of  running  water  (Della  misura  delVacque  correnti,  1628)  which 
repeated  Leonardo  da  Vinci's  law  of  flow,  Sv  =  constant. 

1  fitudes  sur  Leonard  de  Vinci,  Vol.  II,  p.  214. 


144  THE    FORMATION    OF    CLASSICAL   MECHANICS 

Further,  Galileo  related  the  properties  of  the  equilibrium  of  floating 
bodies  to  the  principle  of  virtual  velocities. 

Like  his  contemporaries,  Galileo  also  believed  in  the  horror  vacui 
(resistenza  del  vacuo) .  However,  it  is  reported  that  he  was  very  sur 
prised  to  learn  that  a  newly  constructed  pump,  whose  aspiration  tube 
was  very  long,  could  not  lift  water  higher  than  eighteen  Italian  ells. 
Therefore  he  believed  that  this  height  implied  a  kind  of  ceiling  to  the 
horror  of  the  vacuum.  In  addition,  Galileo  attempted  to  determine  the 
weight  of  air  by  weighing  a  balloon  that  was  filled  with  air,  then  heated 
in  order  to  partially  expel  the  air,  and  weighed  again.  As  Mach  has 
remarked,  it  is  very  true  that  the  heaviness  of  air  and  the  horror  vacui 
were  quite  separate  concepts  before  Pascal's  time.1 

5.  GALILEO  AND  THE  COPERNICAN  SYSTEM. 

We  shall  briefly  summarise  Galileo's  astronomical  work.  By  means 
of  a  lunette  which  he  had  had  constructed  at  Venice  in  1609,  he  disco 
vered  the  satellites  of  Jupiter  on  Jan  7th,  1610  and  observed  that  they 
accompanied  the  planet  in  its  annual  motion.  This  suggested  the  same 
possibility  for  the  Moon  in  relation  to  the  Earth.  On  the  other  hand, 
he  noticed  the  phases  of  Venus  and  the  sunspots,  and  thus  obtained 
proof  of  the  rotation  of  these  two  stars  which  was  of  first  importance 
for  supporting  the  hypothesis  of  the  Earth's  rotation.  Finally  he 
demonstrated  a  libration  in  the  Moon's  longitude.  He  was  forced  to 
retract  his  views  on  the  Earth's  rotation  when  he  was  first  accused  by 
the  Inquisition  in  1615.  Nevertheless  Galileo  hastened  to  publish,  at 
Florence  in  1632,  Four  Dialogues  on  the  two  principal  systems  of  the 
World,  those  of  Copernicus  and  Ptolemy.  (This  in  spite  of  the  fact  that 
he  usually  hesitated  about  printing  his  work  because  of  his  shortage  of 
money ;  even  though  he  was  content  to  distribute  a  few  copies  of  the 
Discorsi  among  his  friends  in  1636.) 

The  three  speakers  that  will  later  appear  in  the  Discorsi,  Simplicio, 
Sagredo,  and  Salviati  also  appear  in  these  dialogues.  Galileo  applied  a 
searching  dialectic  to  the  scholastic  arguments,  here  expressed  by  Simpli 
cio.  For  example,  in  Dialogue  II,  Simplicio  enumerates  the  scholastic 
axioms,  such  as  the  unity  of  the  cause  and  the  unity  of  the  effect,  the 
necessity  of  an  extrinsic  source  for  all  motion,  natural  or  otherwise.  These 
axioms  conflict  with  the  triple  motion  of  Earth  which  Copernicus  has 
suggested.  This  triple  motion  comprises  the  diurnal  motion,  the  annual 
motion  and  the  displacement  of  the  Earth's  axis  parallel  to  itself. 

1  MACH,  AT.,  p.  106. 


GALILEO    AND    TORRICELLI  145 

(Rather  oddly,  Copernicus  had  believed  this  to  be  one  of  the  modes  of 
the  Earth's  motion.)  Salviati  replies  to  this  by  assembling  the  experi 
mental  evidence.  And  if  he  dares  to  contradict  Aristotle,  it  is  because 
the  telescope  has  made  the  eyes  of  the  astronomer  thirty  times  more 
powerful  than  those  of  the  philosopher.  "  Jam  autem  nos,  beneficio 
Telescopii,  tricies  aut  quadragies  propius  quam  Aristoteles  admovemur 
Caelo,  sic  ut  in  eo  plurima  possumus  observare  quae  non  potuit  Aristoteles 
et,  inter  alia,  maculas  istas  in  Sole,  quae  prorsus  ei  fuerunt  invisibiles. 
Ergo  de  Caelo,  deque  Sole,  nos  Aristotele  certius  tractare  possumus.  "  x 

In  his  third  dialogue  Galileo  concludes  that  though  the  copernican 
system  may  be  difficult  to  visualise,  it  is  simple  in  its  effects.  "  Systema 
Copernicanum  intellectu  difficile  et  effectu  facile  est.  " 

It  is  reported  that  this  work  brought  Galileo  a  denunciation  from  the 
Holy  Office,  which  obliged  him  to  renounce  his  copernican  beliefs  and 
to  remain  in  compulsory  residence  at  Arcetri,  near  Florence.  Here  he 
died,  surrounded  by  a  number  of  disciples.  Among  these  was  Torricelli, 
who  had  only  belonged  to  the  circle  for  a  few  months. 

6.    TORRICELLI'S    PRINCIPLE. 

We  know  that  Galileo  had  already  related  the  problem  of  the 
inclined  plane  to  the  principle  of  virtual  work  and  that  he  had  maint 
ained,  in  his  Discorsi,  that  an  ensemble  of  heavy  bodies  could  only 
start  to  move  spontaneously  if  its  centre  of  gravity  came  nearer  to 
the  common  centre  of  heavy  things. 

Torricelli  made  this  remark  precise,  and  raised  it  to  the  status  of 
a  principle,  in  his  treatise  De  Motu  gravium  naturaliter  descendentium 
et  projectorum  (Florence,  1644). 

"  We  shall  lay  down  the  principle  that  two  bodies  connected  together 
cannot  move  spontaneously  unless  their  common  centre  of  gravity 
descends. 

"  Indeed,  when  two  bodies  are  connected  together  in  such  a  way 
that  the  motion  of  one  determines  that  of  the  other,  this  connection 
being  produced  by  means  of  a  balance,  a  pulley  or  any  other  mechanism, 
the  two  bodies  will  behave  as  a  single  one  formed  of  two  parts.  But 
such  a  body  will  never  set  itself  in  motion  unless  its  centre  of  gravity 
falls.  But  if  it  is  made  in  such  a  way  that  its  centre  of  gravity  cannot 
fall,  the  body  will  certainly  remain  at  rest  in  the  position  that  it  occupies. 
From  another  point  of  view,  it  would  move  in  vain  because  it  would 
take  a  horizontal  motion  which  did  not  tend  downwards  in  any  way.  " 

1  We  have  quoted  a  Latin  edition  which  appeared  at  Lyons  in  1641. 

10 


146  THE   FORMATION   OF    CLASSICAL   MECHANICS 

Torricelli  applied  this  principle  to  two  bodies  on  differently  inclined 
planes  and  attached  to  each  other  by  a  weightless  thread.  Similarity, 
he  applied  it  to  the  balance.  All  these  examples  are  instances  of 
indifferent  equilibrium.  If  f  is  the  height  of  the  centre  of  gravity, 
reckoned  algebraically  on  an  ascending  vertical,  Torricelli's  principle 
may  be  written 

(5C-0 

for  all  virtual  displacements  compatible  with  the  constraints.     But  the 
examples  which  he  gave  were  all  of  the  type 

<Jf  =  0. 

Torricelli's  true  merit  is  not  so  much  that  of  having  won  this  principle 
from  Galileo's  mechanics,  but  that  of  having  specified  that  the  verticals 
should  be  treated  as  parallel.  At  the  same  time  he  renounced  the 
scholastic  conception  of  a  common  centre  of  heavy  bodies  at  a  finite 
distance,  where  the  verticals  converged. 

He  writes,  "  This  is  an  objection  that  is  very  common  among 
the  most  thoughtful  authors —  Archimedes  has  made  a  false  hypothesis 
in  regarding  the  threads  that  support  the  two  weights  hung  from  a 
balance  as  being  parallel  to  each  other — in  reality,  the  directions 
of  these  two  threads  meet  at  the  centre  of  the  Earth.  .  .  . 

"  The  foundation  of  mechanics  which  Archimedes  adopted,  namely 
the  parallelism  of  the  threads  of  a  balance,  may  be  deemed  false  when 
the  masses  hung  from  the  balance  are  real  physical  masses,  tending 
towards  the  centre  of  the  Earth.  It  is  not  false  when  these  masses, 
whether  they  be  abstract  or  concrete,  do  not  tend  towards  the  centre 
of  the  Earth,  or  to  any  other  point  near  the  balance,  but  towards  some 
point  which  is  infinitely  distant. 

"  We  shall  continue  to  call  this  point,  towards  which  masses  hung 
from  the  balance  tend,  the  centre  of  the  Earth.  " 

Beneath  these  verbal  precautions,  and  in  spite  of  the  fact  that 
he  did  not  refer  to  the  orders  of  magnitude,  Torricelli's  intention  of 
treating  the  verticals  as  parallels  is  clear.  In  Torricelli's  principle 
the  word  "  descend  "  is  intended  to  indicate  a  tendency  towards  a 
centre  which  is  taken  to  infinity, 

7.  TORRICELLI  AND  THE  MOTION  OF  PROJECTILES. 

Galileo  fully  discussed  the  parabolic  motion  of  a  projectile  which 
was  thrown  horizontally.  Only  in  passing  did  he  remark  that,  if  a 
projectile  was  thrown  obliquely  from  the  point  B  with  a  velocity  equal 


GALILEO   AND    TORRICELLI  147 

and  opposite  to  that  with  which  it  arrived  at  B  after  having  been 
thrown  horizontally  from  A,  it  would  describe  the  same  parabola 
in  the  opposite  direction.  Galileo  made  use  of  this  appeal  to  an 
inverse  return  without  proof.  Moreover,  he  announced  that  the 
greatest  range  for  a  given  velocity,  if  the 
projectile  was  thrown  from  JB,  was  obtained 
when  the  trajectory  at  B  made  half  a  right- 
angle  with  the  horizontal. 

In  this  matter  too,  Torricelli  systematised 
Galileo's  work.  Thus,  in  Book  II  of  his 
De  motu  gravium,  he  considered  a  body  that 
was  projected  obliquely.  He  compounded 
the  uniform  velocity  in  the  direction  of  the 
velocity  of  projection  with  the  accelerated 
motion.  .  ,.,. 

Gassendi  had   studied  the  same  problem  <->  ^ 

as  early  as  1640,  in  a  treatise  Tres  Epistolae 

de  motu  impresso  a  motore  translate.  He  considered  a  body  projected 
upwards  from  the  deck  of  a  ship  in  uniform  motion,  and  showed  that 
the  trajectory  was  a  parabola. 


8.   TORRICELLl'S   EXPERIMENT. 

No  doubt  inspired  by  Galileo's  researches  on  the  resistenza  del  vacua, 
Torricelli  was  lead  to  make  experiments  on  a  column  of  mercury  rather 
than  a  column  of  water.  The  classical  experiment  with  which  his 
name  is  still  associated  was,  however,  accomplished  by  Viviani  in  1643. 


9.    TORRICELLl'S   LAW  FLOW  THROUGH  AN  ORIFICE. 

Torricelli  seems  to  have  been  the  inventor  of  hydrodynamics. 
Thus  he  observed  the  flow  of  a  liquid  through  a  narrow  orifice  near 
the  bottom  of  a  vessel.  Dividing  the  total  duration  of  flow  into  equal 
parts,  he  established  that  the  quantities  of  liquid  caught  by  some 
suitable  receptacle  increased  regularly,  from  the  last  interval  of  time 
to  the  first,  and  that  they  were  proportional  to  the  odd  numbers  taken 
consecutively.  This  analogy  with  the  law  of  falling  bodies  induced 
him  to  investigate  the  height  to  which  the  water  that  flowed  out  of 
the  orifice  could  rise,  if  suitably  directed  upwards.  He  established 
that  this  height  was  always  less  than  that  of  the  liquid  in  the  vessel. 
Moreover,  he  supposed  that  the  stream  would  attain  this  height  if 


148  THE   FORMATION   OF   CLASSICAL  MECHANICS 

the  resistances  did  not  exist.  TorricelE  then  formulated  the  law  that 
the  velocity  of  the  liquid  flowing  out  of  the  orifice  was  proportional 
to  the  square  root  of  the  height  of  the  liquid.  This  statement  was 
obtained  by  analogy  with  the  motion  of  heavy  bodies  and  was  given 
without  proof.  It  attracted  the  attention  of  Newton  and  Varignon, 
and  thus  lies  at  the  bottom  of  the  first  investigations  in  hydrodynamics. 


CHAPTER   THREE 

MERSENNE  (1588-1648) 
AS  AN  INTERNATIONAL  GO-BETWEEN 

IN  MECHANICS 
ROBERVAL  (1602-1675) 


1.  THE  ARRIVAL  OF  FOREIGN  THEORIES  IN  FRANCE.     THE  PART  PLAYED 

BY  MERSENNE. 

In  1634  there  appeared  simultaneously,  in  French,  the  translation 
of  Stevin's  mathematical  work,  P.  Herigone's  Cours  Mathematique 
and  Mersenne's  translation  of  Galileo's  Mechanics. 

Herigone's  Cours  Mathematique  was  inspired  by  Stevin's  work, 
and  took  over  the  proof  concerning  equilibrium  on  an  inclined  plane. 
However,  a  column  of  liquid  was  unhappily  substituted  for  Stevin's 
necklace  of  spheres.  Herigone  also  borrowed  many  things  from 
Guido  Ubaldo  and  the  statics  of  Jordanus  and  his  school — in  particular, 
the  solution  of  the  problem  of  the  inclined  plane.  In  this  he  was 
helped  by  the  Italian  Renaissance  and  the  tradition — which  we  have 
discussed  in  connection  with  Tartaglia — that  honoured  Jordanus* 
contribution  to  statics. 

Though  an  omniverous  reader,  Father  Mersenne  (1588-1648)  did  not 
thereby  arrive  at  a  synthesis  of  his  material.  However,  he  established 
contact  between  the  great  students  of  mechanics,  to  whom  he  was 
continually  posing  questions,  providing  references  and  transmitting 
replies.  His  correspondence  is  like  an  international  review  of  mechanics. 

Mersenne's  Synopsis  mathematica  (1626)  reviewed  the  work  of 
Archimedes,  Luca  Valerio,  Stevin,  Guido  Ubaldo  and  many  others. 
As  early  as  1634  he  translated  Galileo's  Mechanics.  He  told  of  Galileo's 
first  work  on  the  fall  of  bodies  in  Harmonicorum  libri  (1636),  and  added 
to  this  a  traetise  on  mechanics  by  Roberval.  He  also  made  the  work 
of  Benedetto  and  Bernardino  Baldi  known  to  french  students  of  the 
subject.  In  1644  he  published  another  compilation  under  the  title 
Tractatus  mechanicus. 


150  THE   FORMATION   OF   CLASSICAL   MECHANICS 

Muck  original  work  has  only  been  preserved  for  us  in  the  form 
of  letters  to  Mersenne.  In  a  time  when  authors  were  not  liberal  of  refe 
rences,  and  disposed  to  pass  of  their  writings  as  entirely  original,  Mer- 
senne's  self-imposed  task  of  liason  and  dissemination  was  quite  essential. 

2.  ROBERVAL  AND  COMPOUND  MOTION. 

We  cannot  describe  the  work  in  kinematics  in  the  XVIIth  Century, 
for  an  analysis  of  this  would  more  properly  belong  to  a  history  of 
geometry.  Nevertheless,  Roberval's  kinematic  geometry  deserves  a 
special  mention  because  he  was  able  to  solve  the  problem  of  drawing 
tangents  to  different  curves — a  preoccupation  among  geometers  of 
the  time — by  means  of  the  composition  of  velocities. 

The  Treatise  on  compound  motion  was  only  published  by  the  Academic 
des  Sciences  in  1693.  It  was  edited  by  a  gentleman  of  Bordeaux  on 
the  basis  of  Roberval's  lectures,  and  the  latter  confined  his  own  con 
tribution  to  the  addition  of  marginal  notes.  The  essential  principle 
used  in  this  treatise  is  the  following  one. 

"  Using  the  particular  properties  of  the  curved  line  that  will  be 
given  to  you,  examine  the  different  motions  that  a  point  which  describes 
the  line  can  have  in  the  neighbourhood  of  the  point  at  which  you 
wish  to  find  the  tangent.  Of  all  these  component  motions,  take  the 
line  of  the  direction  of  the  compound  motion.  You  will  have  the 
tangent  to  the  curved  line.  " 

For  example,  in  the  curve  described  by  a  point  M  fixed  on  a  circle 
which  rolls  without  sliding  on  a  straight  line,  Roberval  compounds 
an  elementary  translation  of  the  base  with  an  elementary  rotation 
of  the  circle.  This  leads  to  a  direction  of  compound  motion  which 
is  perpendicular  to  the  straight  line  joining  M  to  the  point  of  contact 
of  the  circle  with  the  base.  Roberval  treates  fourteen  examples  in  the 
same  way — for  instance  the  cycloid,  the  conchoid,  Archimedes'  spiral  and 
the  conies,  and  he  succeeds  in  drawing  their  tangents  correctly.  How 
ever,  it  turns  out  that  he  begs  the  question  by  giving  the  components  of 
the  velocity  without  precise  justification.  Descartes  was  to  replace  this 
by  a  method  that  became  that  of  the  instantaneous  centre  of  rotation. 

3.  ROBERVAL'S  TREATISE  ON  MECHANICS. 

In  1650  Roberval  wrote  to  Hevelius,  "  We  have  constructed  a 
new  mechanics  on  the  foundation  already  laid.  Except  for  a  small 
number,  the  ancient  stones  with  which  it  has  previously  been  constructed 
have  been  completely  rejected.  It  consists  of  eight  stages,  corres 
ponding  to  a  similar  number  of  books.  " 


MERSENNE  AND  ROBERVAL  151 

The  Bibliotheque  Nationale  (Paris)  has  a  manuscript  (No.  7226) 
which  is  undoubtedly  an  outline  of  this  work.  Even  though  he  denies 
this,  the  mechanics  which  the  author —  "  in  the  chair  of  Ramus  " 
at  the  College  of  France  — contemplates  is  most  often  inspired  by 
Aristotle  and  the  Italian  Renaissance.  Roberval  claims  to  have 
only  read  Archimedes,  Guido  Ubaldo  and  Luca  Valerio.  But  he  is 
clearly  subject  to  Baldi's  influence.  For  example,  this  is  what  Roberval 
writes  on  the  motion  of  projectiles. 

"  The  violence  of  a  cannon-shot  is  made  up  of  two  impressed 
[motions].  One  is  purely  violent,  arising  from  the  cannon  itself  and 
from  the  powder  which  is  ignited  to  drive  the  shot  along.  The  other 
is  natural,  being  caused  by  the  shot's  own  weight.  Of  the  first  impress 
ion,  the  violence  increases  somewhat  at  some  distance  from  the  cannon 
because  of  the  degrees  acquired  by  the  motion,  which  are  added  to 
the  impression  of  the  powder  before  this  has  decreased  appreciably. 
It  then  happens  that,  since  the  impression  decreases  much  more  in 
itself  than  it  is  added  to  by  the  degrees  of  velocity  acquired,  it  conti 
nually  slows  down  and,  after  a  certain  time,  finishes.  Now  at  the 
beginning  the  line  of  the  direction  of  this  violent  impression  is  directed 
towards  the  place  at  which  the  cannon  points.  Later  it  changes  conti 
nually  and  the  cause  of  this  change  is  the  natural  impression,  that  is, 
the  body's  heaviness  carrying  it  towards  the  centre  of  the  Earth.  For 
the  mixture  of  these  two  impressions,  violent  and  natural,  means  that 
the  shot  does  not  exactly  proceed  along  one  direction  or  the  other.  But 
at  the  beginning  it  almost  entirely  follows  the  violent  one,  which  is, 
without  comparison,  much  greater  than  the  natural  one.  Later  the 
violent  one  disappears  bit  by  bit,  and  so  the  shot  begins  to  descend  a 
curved  line,  and  this  all  the  more  as  the  violent  impression  decreases 
and  the  natural  motion  is  added  to  by  the  degrees  acquired.  " 

4.  ROBERVAL  AND  THE  LAW  OF  COMPOSITION  OF  FORCES. 

Roberval's  first  claim  to  fame  in  statics  is  that  of  having  justified 
the  law  of  the  parallelogram  offerees.  This  he  accomplished  by  starting 
from  the  condition  for  the  equilibrium  of  the  angular  lever.  We  shall 
follow  the  treatise  that  Mersenne  appended  to  his  Harmonicorum  libri. 
Roberval's  work  is  more  modest  than  the  one  to  which  we  have  just 
referred,  and  only  occupies  36  pages. 

We  shall  analyse  Roberval's  demonstrations  instead  of  quoting 
them — their  style  is  heavy  and  artificially  complicated,  while  their  basis 
is  simple. 

In  the  first  place,  Roberval  considers  a  weight  P  suspended  at  B  by 


152 


THE    FORMATION   OF    CLASSICAL   MECHANICS 


two  strings  AB  and  BC.  The  string  A  passes  through  the  fixed  point  A. 
Roberval  sets  out  to  determine  the  traction  Q  which  must  be  applied 
to  the  string  BC  in  order  to  support  the  weight  P.  He  replaces  the 
arm  AB,  whose  length  is  fixed,  by  an  angular  lever  Ap,  Aq,  where  Ap 


Q 


Fig.  49 


is  the  perpendicular  on  the  line  of  action  of  the  weight  P,  and  Aq  is  the 
perpendicular  to  the  string  BC.  The  equilibrium  of  an  angular  lever 
requires  that 

P  =  Aq 

Q~  Ap' 

From  this  the  value  of  Q  is  obtained. 

Roberval  then  applies  this  result  to  the  next  diagram.  Here  QG 
and  CB  are,  respectively,  perpendicular  to  CA  and  QA.  Further,  CF 
and  QD  are  perpendicular  to  the  line  of  action  of  the  weight  A. 

The  weight  A  is  suspended  from  the  two  strings  CA,  QA^  to  which 
are  applied  the  powers  K  and  E.  The  equilibrium  of  the  lever  CF,  CB 

A        CB 

gives  the  ratio  —  =  •—  .     Similarity,  the  lever  ()D,  Q  G  gives  the  ratio 
Jb        CJT 


K~  QD 

"  Therefore  it  is  observed  that  in  both  cases  two  perpendiculars  are 
drawn  from  each  power  —  one  on  the  direction  of  the  weight  and  the 
other  on  the  string  of  the  other  power.  Also  that,  in  the  ratios  of  the 
weight  to  the  powers,  the  weight  is  homologous  to  the  perpendiculars 
falling  on  the  strings  of  the  powers.  Similarly  the  powers  are  homolo 
gous  to  the  perpendiculars  falling  on  the  direction  of  the  weight.  " 


MERSENNE  AND  ROBERVAL 


153 


By  these  purely  geometrical  considerations  Roberval  finally  trans 
forms  the  statement  of  the  preceding  rule  and  arrives  at  the  decompo 
sition  of  the  weight  into  its  two  components  in  the  directions  of  CA 
and  QA. 


"  If,  from  some  point  taken  on  the  line  of  the  direction  of  the  weight, 
the  line  parallel  to  one  of  the  strings  is  drawn  to  the  other  string,  the  sides 
of  the  triangle  thus  formed  will  be  homologous  to  the  weight  and  the  two 


'js  interesting  to  remark  that  Roberval  attempted  to  relate  the 
rule  of  the  composition  of  forces  to  the  principle  of  virtual  work. 

"  In  connection  with  a  weight  suspended  by  two  strings,  we  have 
noticed  a  thing  that  has  given  us  much  pleasure.  This  is  that  when  the 
weight  is  supported  thus  by  two  powers,  it  can  neither  rise  nor  fall 
without  the  reciprocal  proportion  of  the  paths  with  the  weight  and  the 
two  powers  being  changed,  and  this  contrary  to  the  common  order.  .  .  . 

"  If  a  line  AP  is  taken  underneath  A,  in  the  line  of  its  direction,  it 
turns  out  that  if  the  weight  A  falls  as  far  as  P,  drawing  the  strings  with 
it  and  making  the  powers  rise,  the  reciprocal  ratio  of  the  paths  that  the 
powers  travel  in  rising  and  the  path  which  the  weight  travels  in  tailing 
will  be  greater  than  that  of  the  same  weight  and  the  two  powers  taken 
together.  Thus  the  powers  will  be  raised  further  in  the  proportion  that 
the  weight  descends  in  carrying  them  along,  which  is  contrary  to  the 
common  order.  "  .  , 

An  analogous  argument  is  applied  to  the  rising  of  the  weight,  ana 
this  conclusion  follows-  "  Consequently  the  weight  A,  in  remaining  in 
its  place  also  remains  in  the  common  order.  " 


CHAPTER   FOUR 

DESCARTES'   MECHANICS 
PASCAL'S   HYDROSTATICS 


1.  DESCARTES'  STATICS. 

Descartes9  statics  stems  directly  from  the  principle  of  virtual  work, 
which  he  assumed  a  priori.  We  shall  quote  a  letter  from  Descartes  to 
Constantin  Huyghens  dated  October  5th,  1637. 

"  The  invention  of  all  [simple  machines]  is  only  based  on  a  single 
principle,  which  is  that  the  same  force  that  can  lift  a  weight  of,  for 
example,  a  hundred  pounds  to  a  height  of  two  feet,  can  also  lift  one  of 
two  hundred  pounds  to  a  height  of  one  foot,  or  one  of  four  hundred 
pounds  to  a  height  of  half  a  foot,  and  so  on,  however  this  may  be  applied. 

"  And  this  principle  cannot  fail  to  be  accepted  if  it  is  considered 
that  the  effect  should  always  be  proportional  to  the  action  which  is 
needed  to  produce  it.  So  that  if  it  necessary  to  use  the  action  by  which 
a  weight  of  a  hundred  pounds  can  be  lifted  to  a  height  of  two  feet,  in 
order  to  lift  some  weight  to  a  height  of  one  foot,  this  weight  should 
weigh  two  hundred  pounds.  For  it  is  the  same  to  lift  a  hundred  pounds 
to  a  height  of  one  foot,  and  then  again,  to  lift  a  hundred  pounds  to  the 
same  height  of  one  foot  as  to  lift  two  hundred  pounds  to  a  height  of  one 
foot  and  also  the  same  as  to  lift  one  hundred  pounds  to  a  height  of  two 
feet. 

"  Now  the  machines  which  serve  to  make  this  application  of  a  kind 
that  acts  on  a  weight  over  a  great  distance,  and  makes  this  rise  by  a 
smaller  one,  are  the  pulley,  the  inclined  plane,  the  wedge,  the  lathe  or 
turner,  the  screw,  the  lever  and  some  others.  For  if  it  is  not  desired  to 
relate  some,  they  could  be  further  enumerated.  And  if  it  is  desired  to 
relate  them  in  such  a  way,  there  is  no  need  to  put  down  as  many. 

"  If  it  is  desired  to  lift  a  body  F,  of  weight  200  pounds,  to  the  height 
of  the  line  BA,  in  spite  of  the  fact  that  the  force  is  is  only  sufficient  to 
lift  one  hundred  pounds,  it  is  only  necessary  to  drag  or  roll  the  body 


DESCARTES   AND   PASCAL 


155 


F 


Fig.  51 


along  the  inclined  plane  CL4,  which  I  suppose  to  be  twice  as  long  as 
the  line  AB.     For  in  order  to  bring  it  to  the  point  A  by  this  path, 
only  the  force  which  is  necessary  to  make 
a  hundred  pounds  rise  twice  as  high  would 
be  used.  .  .  . 

"  But  to  be  set  against  this  calculation 
is  the  difficulty  there  will  be  in  moving  the 
body  F  along  the  plane  AC  if  this  plane 
had  been  laid  along  the  line  BC,  whose 
parts  I  assume  to  be  equally  distant  from 
the  centre  of  the  Earth.  Since  this  obstruc 

tion  will  be  less  as  the  plane  is  harder,  more  even  and  more  polished,  it  is 
a  fact  that  it  can  only  be  expressed  approximately  and  is  not  very 
considerable.  Further,  there  is  no  need  to  consider  that  the  plane  AC 
should  be  slightly  curved  on  account  of  the  fact  that  the  line  BC  is  a 
part  of  a  circle  which  has  the  same  centre  as  the  Earth  .  .  .  for  this  is 
in  no  way  appreciable.  " 

In  Descartes'  work  on  the  lever,  the  resistance  is  always  a  weight 
hung  from  the  lever,  and  the  power  is  constantly  perpendicular  to  the 
arm  of  the  lever.  G-uido  Ubaldo  had  made  use  of  this  practical  observa 
tion  and  Descartes  followed  him.  We  shall  return  to  the  text. 

"  I  have  postponed  speaking  about  levers  until  the  end  because,  of 
all  the  machines,  used  to  lift  weights,  this  is  the  most  difficult  to  explain. 

"  Consider  this  —  that  while  the  force  which  moves  the  lever  descends 
along  the  whole  semicircle  ABCDE,  although 
the  weight  also  describes  the  semicircle  FGHIK 
it  is  not  lifted  the  whole  length  of  the  line 
FGHIK,  but  only  the  length  of  the  straight 
line  FOK.  So  that  the  proportion  that  the 
force  which  moves  the  weight  must  bear  to 
the  heaviness  of  the  weight  should  not  be 
measured  by  the  proportion  of  the  two  dia 
meters,  but  rather  by  the  proportion  of  the 
greatest  circumference  to  the  smallest  dia 
meter. 

"  Moreover,  consider  that  in  order  to  turn 
the  lever  it  is  by  no  means  necessary  that  the 
force  should  be  as  great  when  the  lever  is  near 
A  or  near  E  as  when  it  is  near  B  or  near  D. 
The  reason  for  this  is  that,  there,  the  weight  rises  less,  as  it  is  easy  to 
see.  And  to  evaluate  exactly  what  this  force  should  be  at  each 
point  of  the  curved  line  ABCDE,  it  is  necessary  to  know  that  it  acts  in 


x^" 


156  THE    FORMATION    OF   CLASSICAL   MECHANICS 

the  same  way  as  if  drew  the  weight  on  a  circular  inclined  plane.  Also 
that  the  inclination  at  each  of  these  points  on  the  circular  plane  should 
be  measured  by  that  of  the  straight  line  which  touches  the  circle  at 
that  point.  " 

Not  only  did  Descartes  assert  the  principle  of  virtual  work  but — and 
in  this  regard  his  priority  is  certain — he  indicated  its  infinitesimal 
character. 

"  The  relative  weight  of  each  body  should  be  measured  by  the  start 
of  the  movement  which  the  power  that  maintains  it  can  produce,  rather 
than  by  the  height  to  which  it  can  rise  after  it  has  fallen  down.  Note 
that  I  say  start  to  fall  and  not  simply  fall,  because  it  is  the  start  of  the 
fall  that  must  be  taken  care  of.  " 

In  passing,  we  recall  Descartes'  contempt  of  his  contemporaries 
and  predecessors.  Naturally  Mersenne  had  drawn  his  attention  to 
Galileo — here  is  Descartes  reply. 

"  And  in  the  first  place,  concerning  Galileo,  I  will  say  to  you  that 
I  have  never  seen  him,  nor  have  I  had  any  communication  with  him, 
and  that  consequently  I  could  not  have  borrowed  anything  from  him. 
Also,  I  see  nothing  in  his  books  that  causes  me  envy,  nor  anything 
approaching  what  I  would  wish  to  call  my  own. 

"  It  seems  to  me  foolish  to  think  of  the  screw  as  a  lever — if  my 
memory  is  correct,  this  is  the  fiction  that  Guido  Ubaldo  used.  " 

To  assert  his  independence  of  Galileo  he  wrote  to  Mersenne  in  the 
following  terms. 

"  As  for  what  Galileo  has  written  on  the  balance  and  the  lever,  it 
explains  the  quod  itafit  rather  well,  but  not  the  cur  ita  fit  as  I  have  done 
with  my  Principle.  "  This  shows  that  Descartes  believed  that  a  prin 
ciple  that  had  been  set  up  overrode  all  other  considerations,  even  experi 
mental  ones.  .  .  . 

In  the  texts  that  we  have  quoted,  Descartes  continually  uses  the 
word  force  to  denote  what  we  now  call  work.  Even  in  his  own  time 
some  misunderstandings  arose,  and  he  was  quick  to  take  offence.  On 
November  15th,  1638,  he  wrote  to  Mersenne  on  this  matter. 

"  At  last  you  have  understood  the  word  force  in  the  sense  that  I  use 
it  when  I  say  that  it  takes  as  much  force  to  lift  a  weight  of  100  pounds 
to  a  height  of  one  foot  as  to  lift  one  of  50  pounds  to  a  height  of  two  feet. 
That  is,  that  as  much  action  or  as  much  effort  is  needed.  "  Descartes 
clarifies  this  later  (September  12th,  1638). 

"  The  force  of  which  I  have  spoken  always  has  two  dimensions  and  is 
not  the  force  which  might  be  applied  at  some  point  to  maintain  a  weight, 
which  always  has  only  one  dimension.  " 

Force  in  Descartes  sense  is  therefore  expressed  by  the  product  pi  of 


DESCARTES    AND    PASCAL  157 

a  weight  and  a  distance  while  the  momenta  in  Galileo's  sense 1  is  express 
ed  by  the  product  pv  of  a  weight  and  a  velocity.  Descartes  formally 
claims  to  have  excluded  consideration  of  the  velocity,  "  which  would 
make  it  necessary  to  attribute  three  dimensions  to  the  force. "  He  adds — 

"  As  for  those  who  say  that  I  should  consider  the  velocity  as  Galileo 
has  done  I  believe,  among  ourselves,  that  they  are  people  who  only 
talk  nonsense  and  that  they  understand  nothing  in  this  matter.  " 

Writing  to  Boswell  in  1646,  Descartes  returned  to  this  theme,  which 
lay  close  to  his  heart. 

"  I  do  not  deny  the  material  truth  of  what  the  students  of  mechanics 
are  accustomed  to  say.  Namely,  that  the  greater  the  velocity  at  the 
end  of  the  long  arm  of  a  lever  is,  in  relation  to  the  velocity  at  the  other 
end,  the  less  force  it  requires  to  be  moved.  But  I  deny  that  the  velocity 
or  the  slowness  are  the  causes  of  this  effect.  " 

Thus  Descartes  rejects  all  connection  between  statics  and  Aristotle's 
dynamics — of  which  traces  subsist  even  in  some  of  Galileo's  concepts. 
Statics  is  made  to  depend  on  a  single  principle,  which  he  asserts  to  be  an 
obvious  reality.  Writing  to  Mersenne  on  September  12th,  1638,  he  said, 
"  It  is  impossible  to  say  anything  good  concerning  the  velocity  without 
having  to  explain  what  heaviness  is,  and,  in  the  end,  the  whole  system 
of  the  World.  " 

With  regard  to  Roberval,  who  had  claimed  Mersenne's  recognition 
of  his  own  priority  in  connection  with  the  postulate  of  statics  that 
Descartes  used,  the  latter  shows  himself  to  be  even  more  contemptuous. 

"  I  have  just  read  your  RobervaPs  Treatise  on  Mechanics,  in  which 
I  learn  that  he  is  a  professor — something  I  did  not  know.  ...  As  for 
his  Treatise,  I  would  be  able  to  find  a  large  number  of  mistakes  in  it 
if  I  wished  to  examine  it  carefully.  But  I  will  say  to  you  that,  on  the 
whole,  he  has  taken  a  great  deal  of  trouble  to  explain  a  thing  that 
is  very  easy,  and  that,  by  means  of  his  explanation,  he  has  made  it 
more  difficult  than  it  naturally  is.  Stevin  showed  the  same  things 
before  him,  and  in  a  much  more  facile  and  general  way.  It  is  true 
that  I  do  not  know  whether  either  of  them  is  correct  in  his  demonstrations, 
for  I  cannot  have  the  patience  to  read  the  whole  of  these  books.  When 
he  claims  to  have  included  something  in  a  Corollary  that  is  the  same 
as  I  have  done  in  my  Writing  on  Statics,  aberrat  toto  Caelo,  he  is  making 
something  that  I  made  a  principle,  a  conclusion,  and  he  talks  of  time 
and  of  velocity  in  places  where  I  talk  of  distance.  This  is  a  very 
serious  mistake,  as  I  have  explained  in  my  earlier  letters.  "  2 

This    haughtiness    had    its    inconveniences.     For    this    refusal    to 

1  Cf.  above,  p.  143. 

2  Letter  to  MERSENNE,  October  llth,  1638. 


158  THE   FORMATION   OF    CLASSICAL   MECHANICS 

read  Roberval  entailed  Descartes'  ignorance  of  the  law  of  the  compo 
sition  of  forces.  In  fact,  the  quantity  pi  had  been  considered  as  a 
measure  of  the  work  done  by  a  weight  by  Jordanus  and  by  Descartes 
contemporaries,  Roberval  and  Herigone. 

We  can  agree  with  Duhem  that  "  Descartes  gave  statics  the  order 
and  the  clarity  which  are  the  very  essence  of  his  method,  but  there 
is  no  truth  in  Descartes'  statics  that  men  had  not  know  before.  Blind 
ed  by  his  prodigious  pride,  he  only  saw  the  error  in  the  work  of  his 
predecessors  and  contemporaries.  "  l 

2.  DESCARTES  AND  THE  FALL  OF  HEAVY  BODIES. 

Descartes  discussed  the  fall  of  bodies  with  Isaac  Beeckman  during 
his  first  stay  in  Holland  (1617-1619).  The  fragment  that  we  are 
going  to  analyse  dates  from  this  time,  but  Descartes  returned  to  the 
subject  in  a  letter  to  Mersenne  dated  November  16th,  1629. 

Descartes  starts  by  recalling  that  a  body  which  falls  from  A  to  B 
and  then  from  B  to  C  travels  much  more  quickly  in  BC  than  in  AB, 

"  for  it  keeps  all  the  impetus  by  means  of 
which  it  moves  along  AB  and  besides,  a  new 
impetus  which  accumulates  in  it  because  of 
the  effect  of  the  gravity,  which  hurries  it 
along  anew  at  each  instant.  "  This  is  the 
scholastic  doctrine  on  the  accumulation  of 
impetus. 

"  The  triangle  ABCDE  shows  the  propor 
tion  in  which  the  velocity  increases. 

"  The  line  1  denotes  the  strength  of  the 
impressed  velocity  at  the  first  moment,  line  2, 
the  strength  of  the  velocity  impressed  at  the 
second  moment,  etc.  .  .  .     Thus  the  triangle 
lg4  ABE  is  formed  and  represents  the  increase 

of  the  velocity  in  the  first  half  of  the  distance 

which  the  body  travels.  As  the  trapezium  BCDE  is  three  times  greater 
than  the  triangle  ABE,  it  follows  that  the  weight  falls  three  times 
more  quickly  from  B  to  C  than  from  A  to  B.  That  is,  that  if  it  falls 
from  A  to  B  in  3  moments,  it  will  fall  from  B  to  C  in  a  single  moment. 
Thus  in  four  moments  its  path  will  be  twice  as  long  as  in  three  ;  in 
twelve,  twice  as  long  as  in  nine  ;  and  so  on.  "  2 

10.  S.,  Vol.  I,  p.  351. 

2  CEuvres  completes,  Vol.  I,  p.  69. 


DESCARTES    AND   PASCAL 


159 


Therefore  Descartes,  like  Galileo  in  1604,  assumed  the  law  v=ks. 
Before  he  developed  his  own  analytical  geometry,  he  used  the  geome 
trical  representation  of  uniformly  varying  quantities  that  was  due  to 
Oresme.  Descartes  called  the  measure  of  such  a  quantity  the  augmen- 
tatio  velocitatis.  But  he  confused  the  augmentatio  along  AB  with  the  mean 
velocity  along  AB,  which  lead  to  a  conclusion  that  was  not  only  incorrect, 
but  also  in  contradiction  with  the  law  from  which  he  had  started. 

Isaac  Beeckman's  Journal,  which  was  used  by  Adam  and  P.  Tannery 
in  their  edition  of  Descartes'  works,  contains  further  details  of  these 
discussions.1 

Beeckman  assumes  the  correct  law  v=kt  and  correctly  deduces 
from  it  the  law  of  distances.  If  AD  represents  a  duration  of  one  hour, 
the  distance  travelled  is  represented  by  the  triangle  ADE.  In  two 
hours  the  distance  is  represented  by  the  triangle  ADC.  The  ratio 
of  the  areas,  and  therefore  that  of  the  distances,  is  therefore  the  squared 
ratio  of  the  times. 

Beeckman  makes  use  of  the  method  of  indivisibles  in  order  to 
justify  this  result.  "  If,  during  the  first  moment  of  time,  the  body 
has  travelled  a  moment  of  distance  AIRS,  during  the  first  two  moments 
of  time  it  will  have  travelled  3  moments  of 
distance,  represented  by  the  figure  AJTURS. 
The  distance  travelled  in  any  time  whatever 
is  therefore  represented  by  the  corresponding 
triangle  supplemented  by  the  small  triangles 
ASR,  RUT,  etc. . . .  which  are  equal  to  each 
other.  But  these  equal  triangles  added  in  this 
way  are  smaller  as  the  moments  of  distance 
are  smaller.  Therefore  these  added  areas 
will  be  of  zero  magnitude  when  it  is  supposed 
that  the  moment  is  of  magnitude  zero.  It 
follows  that  the  distance  which  the  thing 
falls  in  one  hour  is  to  the  distance  through 
which  it  falls  in  two  hours  as  the  triangle 
ADE  is  to  the  triangle  ACB.  " 

The  two  propositions  which  Dominic  Soto  has  stated  are  thus 
linked  with  each  other  by  the  bond  of  indivisibles. 

Beeckman  ascribed  even  this  argument  to  Descartes.  "  Haec  ita  de- 
monstravit  Mr.  Peron.  " 

Unfortunately  Beeckman  did  not  persist  in  this  point  of  view. 
In  another  writing 2  he  went  back  to  the  law  v  =  ks  and  repeated 

1  (Euvres  completes,  Vol.  X,  p.  58. 

2  Ibid.,  Vol.  X,  p.  75. 


Fig.  54 


160 


THE    FORMATION   OF   CLASSICAL   MECHANICS 


the  very  same  error  as  Descartes  in  the  evaluation  of  the  mean  velocity. 
In  spite  of  the  efforts  which  Duhem  made  to  elucidate  them,1  these 
essays  remain  somewhat  confused.  In  this  matter  of  the  fall  of  heavy 
bodies,  it  remains  that  Descartes'  contribution  was  not  lasting  and  much 
less  than  that  of  Galileo,  whose  progress  from  1604  to  1638  was  conti 
nuous.  Moreover,  Galileo  had  a  respect  for  observation,  which  Des 
cartes  eschewed. 

3.  DESCARTES   AND    THE   CONSERVATION   OF  QUANTITIES   OF   MOTION. 

As  early  as  1629  Descartes,  writing  to  Mersenne,  was  categorical 
on  the  indestructibility  of  motion.  "  I  suppose  that  the  motion  that 
is  once  impressed  on  a  body  remains  there  forever  if  it  is  not  destroyed 
by  some  other  means.  In  other  words,  that  something  which  has  start 
ed  to  move  in  the  vacuum  will  move  indefinitely  and  with  the  same 
velocity.  " 

In  his  Dioptrics  Descartes  fell  back  on  a  mechanical  model  to  ex 
plain  the  laws  of  reflection.  A  ball  impelled  from  A  to  B  bounces  off 
the  earth  CBE.  He  explicitly  neglects  "  the  heaviness,  the  size  and 


Fig.  55 


the  shape  "  of  the  ball,  and  supposed  the  earth  to  be  "  perfectly  hard 
and  flat.  "  He  asserts  that  on  meeting  the  earth  the  ball  is  reflected, 
and  the  "  determination  to  tend  to  B  which  it  had  "  is  modified  "  with 
out  there  being  any  other  alteration  of  the  force  of  its  motion  than  this.  " 
In  this  connection,  but  in  passing,  he  denied  the  theory  of  intermediate 
rest,  which  was  dear  to  the  hearts  of  some  of  the  Schoolmen.  He 

1  DUHEM,  Etudes  sur  Leonard  de  Kinci,  Vol.  Ill,  p.  566. 


DESCARTES    AND    PASCAL  161 

deemed  "  the  determination  to  move  towards  some  direction,  like 
the  movement,  to  be  divided  into  all  the  parts  of  which  it  can  be  im 
agined  that  it  is  composed.  "  The  ball  is  thus  animated  by  two  "  deter 
minations.  "  One  makes  it  descend  and  the  other  makes  it  travel 
horizontally.  The  impact  with  the  ground  can  disturb  the  first  but 
can  have  no  effect  on  the  second.  Combining  these  principles  with 
that  of  the  conservation  of  the  force  of  the  motion  of  the  ball,  Descartes 
explained  the  laws  of  reflection. 

In  his  Principles  (1644),  Descartes  reasserts  the  conservation  of 
motion  in  a  very  detailed  way,  making  it  part  of  a  metaphysical  system. 

"  God  in  his  omnipotence  has  created  matter  together  with  the 
motion  and  the  rest  of  its  parts,  and  with  his  day-to-day  interference, 
he  keeps  as  much  motion  and  rest  in  the  Universe  now  as  he  put  there 
when  he  created  it.  ..." 

By  motion,  Descartes  understands  what  we  now  call  quantity  of 
motion,  however  precise  his  ideas  on  mass  may  be. 

64  When  a  part  of  matter  moves  twice  as  quickly  as  another  that 
is  twice  as  large,  we  ought  to  think  that  there  is  as  much  motion  in 
the  smaller  part  as  in  the  larger.  And  that  each  time  the  motion  of 
one  part  decreases,  that  of  some  other  part  is  increased  proportionally.  " 

Further,  Descartes  asserts  the  relativity  of  motion. 

"  We  would  not  be  able  to  understand  that  the  body  AB  is  moved 
from  the  neighbourhood  of  the  body  CD  if  we  did  not  also  know  that 
the  body  CD  is  moved  from  the  neighbourhood  of  the  body  AB.  No 
difficulty  is  created  by  saying  that  there  is  as  much  motion  in  one  as 
in  the  other.  " 

Moreover,  he  distinguishes  between  the  proper  motion  of  a  body 
and  "  the  infinity  of  motions  in  which  it  can  participate  because  it 
is  part  of  other  bodies  which  move  differently.  "  In  order  to  illustrate 
this,  he  gives  the  example  of  the  watch  of  a  sailor  that  takes  part 
in  the  motion  of  his  vessel. 

Again,  Descartes  affirmed  that  the  motion  is  conserved  in  a  straight  line : 

"  Each  part  of  nature,  in  its  detail,  never  tends  to  move  along  curved 
lines,  but  along  straight  lines.  This  rule . . .  results  from  the  fact  that 
God  is  immutable  and  conserves  motion  in  nature  by  a  very  simple 
operation  ;  for  he  does  not  conserve  it  as  it  might  have  been  some  time 
previously,  but  as  it  is  at  the  precise  moment  he  conserves  it.  "  And 
Descartes  here  recalls  the  motion  of  a  stone  in  a  sling  ;  he  points  out 
that  we  cannot  "  conceive  any  curvature  in  the  stone,  "  of  that  we  are 
"  assured  by  experience  "  for  the  stone  leaves  "  straight  from  the  sling. . . ; 
which  makes  manifest  to  us  that  any  body  that  is  moved  in  a  circle 
tends  unceasingly  to  recede  from  the  centre  of  the  circle  it  describes  ; 
11 


162  THE   FORMATION   OF   CLASSICAL  MECHANICS 

and  we  can  even  feel  this  with  our  hand  while  we  turn  the  stone  in  the 
sling,  because  it  pulls  and  makes  the  string  taut  in  its  effort  to  recede 
directly  from  our  hand.  " 

4.  DESCARTES  AND  THE  IMPACT  OF  BODIES. 

Descartes  formulated  the  following  rules  for  the  impact  of  bodies. 

1)  If  two  equal  bodies  impinge  on  one  another  with  equal  velocity, 
they  recoil,  each  with  its  own  velocity. 

2)  If  one  of  the  two  is  greater  than  the  other,  and  the  velocities 
equal,  the  lesser  alone  will  recoil,  and  both  will  move  in  the  same 
direction  with  the  velocity  they  possessed  before  impact. 

3)  If  two  equal  bodies  impinge  on  one  another  with  unequal  vel 
ocities,  the  slower  will  be  carried  along  in  such  a  way  that  their  common 
velocity  will  be  equal  to  half  the  sum  of  the  velocities  they  possessed 
before  impact. 

4)  If  one  of  the  two  bodies  is  at  rest  and  another  impinges  on  it, 
this  latter  will  recoil  without  communicating  any  motion  to  it. 

5)  If  a  body  at  rest  is  impinged  on  by  a  greater  body,  it  will  be 
carried  along  and  both  will  move  in  the  same  direction  with  a  velocity 
which  will  be  to  that  of  the  impinging  body  as  the  mass  of  the  latter  is 
to  the  sum  of  the  masses  of  each  body. 

6)  If  a  body  C  is  at  rest  and  is  hit  by  an  equal  body  J5,  the  latter 
will  push  C  along  and,  at  the  same  time,  C  will  reflect  B.     If  B  has  a 
velocity  4  it  gives  a  velocity  1  to  C  and  itself  moves  backwards  with 
velocity  3. 

This,  as  an  example,  is  how  Descartes  justifies  this  rule. 

"  It  is  necessary  that  either  B  will  push  C  along  and  not  be  reflected, 
and  thus  transfer  2  units  of  its  own  motion  to  C  ;  or  that  it  will  be  reflec 
ted  without  pushing  C  along  and,  as  a  consequence,  that  it  will  retain 
these  2  units  of  velocity  together  with  the  2  that  cannot  be  lost  to  it ; 
or,  further,  that  it  will  be  reflected  and  retain  a  part  of  these  two 
units  and  that  it  will  push  C  along  by  transferring  the  other  part. 
It  is  clear  that,  since  the  bodies  are  equal  and,  consequently,  that 
there  is  no  reason  why  B  should  be  reflected  rather  than  that  it  should 
push  C  along,  these  two  effects  will  be  equally  divided.  That  is  to 
say  that  B  will  transfer  to  C  one  of  these  2  units  of  velocity  and  will 
be  reflected  with  the  other.  " 

7)  Descartes  also  formulated  a  seventh  rule  relating  to  two  unequal 
bodies   travelling  in   the   same   direction. 

We  remark  that  Descartes'  guiding  idea  was  the  conservation  of 
the  quantity  of  motion,  m\v  ,  in  absolute  value.  This  idea  was  to  persist 


DESCARTES   AND   PASCAL  163 

among  the  Cartesians  until  the  resolution  of  the  controversy  about 
living  forces  that  we  shall  come  to  much  later. 

Nearly  all  Descartes'  rules  on  impact  are  experimentally  incorrect. 
We  remark  that  he  may  have  suspected  this,  without  being  very  much 
disturbed,  when  he  said — 

"  It  often  happens  that,  at  first,  the  observations  seem  to  be  at 
variance  with  the  rules  I  have  just  described.  But  the  reason  for 
this  is  clear.  For  the  rules  presuppose  that  the  two  bodies  B  and  C 
are  perfectly  hard  and  so  separated  from  all  others  that  there  is  no 
other  near  them  which  can  help  or  hinder  their  motion.  We  see  nothing 
of  this  kind  in  the  World.  " 

Jouguet  has  a  very  legitimate  comment  to  make  on  the  preceding 
declaration.  *6  This  passage  is  very  characteristic  of  Descartes' 
thought.  He  could  observe  nature  and  argue  accurately  from  his 
laws  as  well  as  any  other.  But  he  had  the  pretension  of  rebuilding 
everything  in  a  rational  way  according  to  the  principles  of  his  philosophy. 
He  considered  that  the  source  of  certainty  lay  in  thought  alone.  It 
is  known  that  he  did  not  wish  to  assume  the  principles  that  were  accepted 
in  geometry  and  physics — further,  by  an  exaggeration  of  his  system, 
he  came  to  neglect  observation.  "  * 

5.  THE   DISCUSSION   BETWEEN   DESCARTES   AND   ROBERVAL   ON  THE 
CENTRE  OF  AGITATION. 

In  his  Exercitationes  we  know  that  Bernardino  Baldi  had  introduced 
a  distinction  between  the  centre  of  gravity  and  the  centre  of  violence, 
or  the  centre  of  accidental  gravity.2 

Mersenne,  who  had  read  Baldi's  work,  suggested  to  geometers 
that  they  should  search  for  a  solid  that  would  have  the  same  period 
of  oscillation  as  a  simple  pendulum  of  given  length.  Descartes  replied 
to  him  in  a  letter  dated  March  2nd,  1646.3 

"  [The]  point  of  your  letter — which  I  do  not  wish  to  postpone 
answering — is  the  question  concerning  the  size  that  each  body  that 
is  hung  in  the  air  by  one  of  its  extremities  should  have,  whatever  its 
shape  may  be,  in  order  that  it  should  carry  out  its  comings  and  goings 
equally  with  those  of  a  lead  hung  by  a  thread  of  given  length. ,  .  .  The 
general  rule  that  I  give  in  this  connection  is  the  following  one.  Just 
as  there  is  a  centre  of  gravity  in  all  heavy  bodies,  so  there  is  also  a 
centre  of  their  agitation  in  the  same  bodies,  when  these  are  hung  from  some 

1  L.  AT.,  Vol.  I,  p.  90. 

2  See  above,  p.  106. 

8  CEuvres  completes,  Vol.  IV,  pp.  362-364. 


164  THE   FORMATION   OF   CLASSICAL  MECHANICS 

part  and  move.  Also,  that  all  those  bodies  in  which  this  centre  of 
oscillation  is  equally  distant  from  the  point  from  which  they  are  sus 
pended,  execute  their  comings  and  goings  in  equal  times,  provided 
that  the  change  of  this  proportion  that  can  be  produced  by  the  air 
is  excepted.  " 

He  returned  to  this  question  on  March  30th,   1646.1 

"  In  the  first  place,  since  the  centre  of  gravity  is  so  situated  in 
the  middle  of  a  heavy  body,  the  action  of  any  part  of  the  body,  which 
could,  by  its  weight,  divert  this  centre  from  the  line  along  which  it 
falls,  is  prevented  by  another  part  which  is  opposed  to  the  first  and 
which  has  just  as  much  force.  From  this  it  follows  that,  in  descent, 
the  centre  of  gravity  always  moves  along  the  same  line  as  it  would 
take  if  it  were  alone,  and  if  all  the  other  parts  of  which  it  is  the  centre 
were  taken  away.  Similarly,  what  I  call  the  centre  of  agitation  of  a 
suspended  body  is  the  point  to  which  the  different  agitations  of  all 
the  other  parts  of  the  body  are  related,  so  that  the  force  which  each 
part  has  to  make  itself  move  more  or  less  quickly  than  it  does  is  pre 
vented  by  that  of  another  which  is  opposed  to  it.  From  which  it 
also  follows,  ex  definitione,  that  this  centre  of  agitation  will  move  about 
the  axle  from  which  the  body  is  suspended  with  the  same  velocity 
that  it  would  have  if  the  remainder  of  the  body  of  which  it  is  a  part 
were  taken  away  and,  as  a  consequence,  the  same  velocity  as  a  lead 
hung  from  a  thread  at  the  same  distance  from  the  axle.  " 

Here  Descartes  raised  an  analogy  to  the  status  of  a  principle, 
and  drew  all  the  logical  consequences  from  it  with  his  accustomed 
vigour  and  clarity.  But  he  did  not  confine  himself  to  this.  He  tried 
to  determine  the  position  of  the  centre  of  agitation  by  forming  the 
quantity  mv  for  each  element,  or  the  proportional  quantity  mr  (where 
r  is  the  distance  from  the  axis).  But  he  took  no  account  of  the  direction 
of  these  velocities — he  always  considered  v,  not  v — and  related  every 
thing  to  a  plane  passing  through  the  centre  of  gravity  and  the  axis 
of  rotation. 

Roberval  pointed  out  this  error  in  a  letter  to  Cavendish  dated  May, 
1646.2  "  The  defect  of  [Descartes']  argument  is  that  he  considers 
only  the  agitation  of  the  parts  of  the  agitated  body,  forgetting  the  direc 
tion  of  the  agitation  of  each  of  those  parts.  For  the  centre  of  gravity 
is  the  cause  of  its  reciprocation  from  left  to  right.  " 

Descartes  replies  3  that  Roberval  was  mistaken  "  in  thinking  that 
the  centre  of  gravity  contributes  anything  to  the  measure  of  its  vibra- 

1  CEuvres  completes,  Vol.  IV,  pp.  379-388. 

2  Ibid.,  p.  400. 

3  Ibid.,  p.  432. 


DESCARTES    AND   PASCAL 


165 


tions  beyond  what  the  centre  of  agitation  does.  For  the  word  centre 
of  gravity  is  relative  to  bodies  that  move  freely  or  which  do  not  move 
in  any  way.  For  those  which  move  about  an  axle  to  which  they  are 
attached,  there  is  no  centre  of  gravity  with  respect  to  that  position 
and  the  motion  has  only  a  centre  of  agitation.  " 


Fig.  56 


Without  delaying  ourselves  further  with  this  controversy  in  which 
Descartes  appears  as  his  usual  peremptory  self,  we  shall  confine  our 
selves  to  the  question  of  a  plane  figure  oscillating  about  an  axis  through 
O.1  As  we  have  already  indicated,  Descartes'  calculation  starts  by 
bringing  each  element  of  the  figure,  M,  to  M!  along  the  arc  MM'  of 
centre  0.  The  motion  is  then  supposed  to  correspond  to  a  quantity 
of  motion  applied  perpendicularly  to  OG  at  M'.  On  the  other  hand, 
Roberval  supposes  that  nw  is  applied  at  the  point  I  on  OG.  Only  the 
component  normal  to  OG  matters,  since  the  other  is  nullified  by  the 
fixity  of  the  support.  Thus  Roberval  arrives  at  a  correct  determination 
of  the  centre  of  oscillation  0'  in  the  case  of  a  sector  of  a  circle  oscillating 
about  an  axis  passing  through  the  centre. 

Morsenne's  correspondence  has  established  that  Huyghens  was  in 
volved  in  the  same  question  as  early  as  1646,  at  the  age  of  seventeen. 
At  the  beginning,  his  attitude  to  the  problem  was  determined  by  the 
Cartesian  discipline  and  he  did  not  emancipate  himself  from  this  until 
much  later,  when  he  solved  the  problem  of  the  centre  of  oscillation  in 
his  Horologium  oscillatorium  (1673)  by  appealing  to  the  principle  of  living 
forces. 

1  C/.  JotrcuET,  L.  M.,  Vol.  I,  p.  158. 


166  THE   FORMATION   OF   CLASSICAL  MECHANICS 

6.   THE   QUARREL  ABOUT   GEOSTATICS. 

We  are  now  going  to  say  a  little  about  a  controversy  which  has,  at 
least,  the  interest  of  showing  that  traces  of  Scholasticism  remained 
even  in  the  most  distinguished  minds  of  the  XVIIth  Century. 

In  1635  Jean  de  Beaugrand  announced  to  Galileo,  Cavalieri,  Castelli 
and  those  interested  in  mechanics  in  France,  that  he  had  found  the  law 
which  determined  how  the  weight  of  a  body  varied  with  its  distance 
from  the  centre  of  the  Earth—  the  weight  was  proportional  to  the 
distance. 

This  result  appeared  in  1636  in  his  Geostatics. 

In  a  letter  to  Mersenne,  Descartes  denied  this  in  the  following  terms. 

"  Although  I  have  seen  many  scpiarings  of  the  circle,  perpetual 
motions  and  many  other  would-be  demonstrations  which  were  false, 
I  can  nevertheless  say  that  I  have  never  seen  so  many  errors  united  in 

one  single  proposition Thus  I  can  say  in  conclusion  that  what 

this  book  on  Geostatics  contains  is  so  irrelevant,  so  ridiculous  and 
mistaken  that  I  wonder  that  any  honest  man  has  ever  deigned  to  take 
the  trouble  of  reading  it.  I  would  be  ashamed  of  that  which  I  have 
taken  in  recording  my  feeling  in  this  letter  if  I  had  not  done  so  at  your 
request.  " 

In  May,  1636,  Fermat  formulated  a  proposition  which  he  called 
Propositio  Geostatica  and  which  he  expressed  in  the  following  way. 

Let  B  be  the  centre  of  the  Earth,  BA  a  terrestrial  radius  and  J5C  a 
part  of  the  opposite  radius.  Consider  two  bodies  A^  and  C  which  are 
placed  at  A  and  C.  If  the  weight  A  is  to  the  weight  C  as  JBC  is  to  JEL4, 
the  two  bodies  are  in  e<pzilibrium. 


Fig.  57 

He  adds,  "  It  is  very  easy  to  demonstrate  this  result  by  following  in 
the  steps  of  Archimedes.  " 

Given  this,  Fermat  deduces  the  following  result  from  his  geostatic 
hypothesis. 


// 

-e 


* 


Fig.  58 


DESCARTES   AND   PASCAL  167 

44  Wherever  a  body  JV  is  placed  between  B  and  A,  if  the  proportion 
of  AB  to  BN  is  equal  to  the  proportion  of  the  weight  JV  to  the  power  R 
which  is  applied  at  A,  the  weight  JV  will  be  kept  in  equilibrium  by  the 
power.  Therefore  the  nearer  a  body  approaches  the  centre  of  the  Earth, 
the  smaller  is  the  power  at  A  that  is  necessary  to  maintain  it  in  equili 
brium.  This,  errors  apart,  coincides  with  Beaugrand's  geostatic  pro 
position.  "  x 

Fermat  gave  the  following  explanation  to  Mersenne,  who  had  not 
indicated  his  accord  with  this  strange  proposition  in  which  Fermat  had 
applied  the  laws  of  a  lever  to  longitudinal  forces.2 

44  Every  body,  in  whatever  place  except  the  centre  of  the  Earth  it 
may  be,  and  taken  by  itself  and  absolutely,  always  weighs  the  same.  .  .  . 
In  my  Proposition  I  never  consider  the  body  by  itself,  but  only  in 
relation  to  a  lever,  and  thus  there  is  nothing  in  the  conclusions  which  is 
not  included  in  the  premisses. 

44  Let  A  be  the  centre  of  the  Earth,  and  let  the  body  E  be  at  the 
point  E  and  the  point  JV  be  in  the  surface  or  somewhere  else  that  is 
further  away  from  the  centre  than  the  body  E.  I  do  not 
say  that  E  weighs  less  when  it  is  at  E  than  when  it  is  at  JV. 
But  I  do  say  that  if  the  body  is  suspended  from  the  point  JV 
by  the  thread  JVE,  this  force  at  the  point  JV  will  support  it 
more  easily  than  if  it  were  nearer  to  the  said  force,  and  this 
in  the  proportion  that  I  have  indicated  to  you.  " 

Fermat  was  legitimately  attacked  by  Et.  Pascal  and 
Roberval.  Descartes  also  condemned  him,  and  in  this  con 
nection,  made  clear  his  own  ideas  on  heaviness  in  a  Note 
of  July  13th,  1638. 

64  It  is  necessary  to  decide  what  is  meant  by  absolute  Fig.  59 
heaviness.  Most  people  understand  it  as  an  internal  pro 
perty  or  quality  in  each  of  those  bodies  that  are  called  heavy,  which 
makes  these  bodies  tend  towards  the  centre  of  the  Earth.  According 
to  some,  this  property  depends  on  the  shape,  according  to  others,  only 
on  the  material.  Now  according  to  these  two  beliefs,  of  which  the  first 
is  most  common  in  the  Schools  and  the  second  most  often  accepted  by 
those  who  can  understand  something  out  of  the  ordinary,  it  is  clear 
that  the  absolute  heaviness  of  bodies  is  always  the  same  and  that  it 
does  not  change  in  any  way  because  of  their  different  distances  from  the 
centre  of  the  Earth. 

44  There  is  also  a  third  belief — that  of  those  who  believe  that  there 
is  no  heaviness  which  is  not  relative,  and  that  the  force  or  property 

1  (Euvres  completes  de  Fermat,  Vol.  II,  p.  6. 

2  Ibid.,  p.  17,  Letter  of  Fermat  to  Mersenne,  24th  June  1636. 


168  THE   FORMATION   OF    CLASSICAL   MECHANICS 

which  makes  the  bodies  that  we  call  heavy  descend  does  not  lie  in  the 
bodies  themselves,  but  in  the  centre  of  the  Earth  or  in  all  its  mass,  and 

that  this  attracts  them  towards  itself  as  a  magnet  attracts  iron 

And  according  to  these,  just  as  the  magnet  and  all  other  natural  agents 
which  have  a  sphere  of  activity  are  always  more  effective  at  small  than 
at  great  distances,  so  it  should  be  said  that  the  same  body  weighs  more 
when  it  is  closer  to  the  centre  of  the  Earth. 

"  For  myself,  I  understand  the  true  nature  of  heaviness  in  a  sense 
that  is  very  different  from  these  three.  .  .  .  But  all  that  I  can  say  [here] 
is  that  by  this  I  do  not  add  anything  to  the  clarification  of  the  proposed 
question,  x  except  that  it  is  a  purely  factual  one.  That  is  to  say  that  it 
can  only  be  settled  by  a  man  in  so  far  as  he  can  make  certain  observa 
tions,  and  also  that  from  the  observations  that  are  made  here  in  our  air 
it  is  not  possible  to  know  what  there  might  be  much  lower  down,  towards 
the  centre  of  the  Earth,  or  much  higher,  among  the  clouds.  Because 
if  there  is  a  decrease  or  increase  of  heaviness,  it  is  not  obvious  that  it 
follows  the  same  proportion  throughout.  " 

Descartes  was  of  the  opinion  that  observation  seemed  to  show  that 
heaviness  decreased  as  a  weight  was  separated  from  the  centre  of  the 
Earth.  He  gave  strange  evidence  for  this,  such  as  "  the  flight  of  birds, 
the  paper  dragons  that  children  fly  and  the  balls  of  pieces  of  artillery 
that  are  fired  directly  towards  the  zenith  and  appear  not  to  fall  down 
again.  "  Another  piece  of  evidence  was  that  "  since  the  planets  which 
do  not  have  light  inside  themselves,  like  the  Moon,  Venus  and  Mercury, 
are  probably  bodies  of  the  same  kind  as  the  Earth,  and  since  the  skies 
are  liquid  as  nearly  all  Astronomers  of  this  Century  believe,  it  seems  that 
these  planets  would  be  heavy  things  and  would  fall  towards  the  Earth 
if  their  great  separation  from  it  had  not  removed  this  inclination.  " 

Returning  to  observation  in  the  neighbourhood  of  the  Earth,  Des 
cartes  considered  the  absolute  heaviness  as  being  practically  constant. 
"  If  this  equality  in  the  absolute  heaviness  is  supposed,  it  can  be  shown 
that  the  relative  heaviness  of  all  hard  bodies,  considered  in  the  free  air 
without  any  support,  is  somewhat  less  when  they  are  near  the  centre 
of  the  Earth  than  when  they  are  separated  from  it,  though  it  may  not 
be  the  same  for  liquid  bodies.  On  the  contrary,  if  it  is  supposed  that 
two  equal  weights  are  opposed  to  each  other  on  a  perfectly  accurate 
balance,  when  the  arms  of  the  balance  are  not  parallel  to  the  horizon, 
that  one  of  the  two  bodies  which  is  nearer  the  centre  of  the  Earth  will 
weigh  more  precisely  to  the  extent  that  it  is  closer  to  it.  To  leave  the 
example  of  the  balance,  it  also  follows  that  of  the  equal  parts  of  the 

1  The  question  is  that  of  knowing  whether  a  body  weighs  more  or  less  when  it  is 
near  the  centre  of  the  Earth  than  when  it  is  further  away. 


DESCARTES    AND   PASCAL  169 

same  body,  the  highest  parts  weigh  more  than  than  the  lowest  ones  to 
the  extent  that  they  are  further  separated  from  the  centre  of  the  Earth, 
so  that  the  centre  of  gravity  cannot  be  fixed  in  any  body,  even  a  spher 
ical  one.  " 

As  a  general  rule  the  convergence  of  the  verticals  renders  consider 
ation  of  the  centre  of  gravity  illusory.  Such  was  Descartes'  conclusion. 
He  also  assumed,  as  Guido  Ubaldo  had  already  done,  the  law  of  attrac- 

k 
tion  -  (inversely  proportional  to  the  distance). 

This  conclusion  was  important  and,  if  one  is  to  [believe  Duhem,1  it 
stemmed  from  the  Italian  School — from  Torricelli  through  the  agency 
of  Castelli.  However,  the  Beaugrand-Fermat  law  of  attraction  kr  (pro 
portional  to  the  distance)  allowed  the  existence  of  a  fixed  centre  of 
gravity  in  a  body.  This  had  been  shown  by  P.  Saccheri  in  his  Neo 
Stattica. 


7.  PASCAL'S  HYDROSTATICS. 

Mersenne  had  advertised  Torricelli's  experiment  in  France  as  early 
as  1644.  Pascal  set  about  repeating  the  experiment  with  the  collabo 
ration  of  Petit  and  convinced  himself,  by  this  means,  of  the  possibility 
of  a  vacuum,  "  which  Nature  does  not  avoid  with  as  much  horror  as 
many  imagine.  "  In  1647  Pascal  published  New  Experiments  concerning 
the  Vacuum.  He  gave  full  details  of  his  plan  for  his  great  experiment  in 
a  letter  to  Perier  dated  November  15th,  1647,  and  the  experiment  itself 
was  completed  at  the  Puy  de  Dome — a  mountain  in  Central  France — on 
September  19th,  1648.  The  account  of  the  experiment  was  published 
in  October  of  the  same  year.  The  principal  result  was  that  the  difference 
of  level  of  mercury  columns  separated  by  a  height  of  500  toises  was 
3  pouces,  one  and  a  half  lines. 

The  Treatise  on  the  Equilibrium  of  Liquids  and  the  Heaviness  of  the 
Mass  of  Air  appeared  in  1663. 

In  this  work  Pascal  established  that  liquids  "  weigh  "  according  to 
their  height,  and  that  in  this  respect  a  vessel  of  ten  pounds  capacity 
was  equivalent  to  a  vessel  of  one  ounce  capacity  if  both  heights  were 
the  same.  From  this  Pascal  directly  obtained  the  principle  of  the 
hydraulic  press. 

"  A  vessel  full  of  water  is  a  new  principle  in  mechanics,  and  a  new 
machine  for  multiplying  forces  to  whatever  degree  might  be  desired.  " 

Pascal  immediately  relates  this  principle  to  that  of  virtual  work. 

1  0.  S.,  Vol.  II,  p.  183. 


170 


THE   FORMATION   OF   CLASSICAL   MECHANICS 


**  And  it  is  wonderful  that  in  this  new  machine  there  is  encountered  the 
same  constant  order  that  is  found  in  all  the  old  ones,  namely  the  lever, 
the  windlass,  the  endless  screw,  .  .  .  which  is  that  the  path  increases  in 
the  same  proportion  as  the  force.  ...  It  is  clear  that  it  is  the  same 
thing  to  make  a  hundred  pounds  travel  a  path  of  one  pouce  as  to  make 
one  pound  travel  a  path  of  a  hundred  pouces.  " 

It  is  in  this  connection  that  Pascal  defines  the  pressure — the  water 
beneath  both  pistons  of  a  hydraulic  press  is  equally  compressed. 

In  Chapter  II  of  the  Treatise  on  the  Equilibrium  of  Liquids  there  is 
mention  of  a  "  small  treatise  on  mechanics, "  now  lost,  in  which  Pascal 
had  given  "  the  reason  for  all  the  multiplications  of  forces  which  are 
found  in  all  the  instruments  of  mechanics  so  far  invented-  " 

This  principle  does  not  seem  to  have  differed  from  Torricelli's  prin 
ciple  which  Pascal  used,  without  quotation,  in  hydrostatics. 


Fig.  60 


"  I  take  it  as  a  principle  that  a  body  never  moves  because  of  its 
weight  unless  the  centre  of  gravity  descends.  From  this  I  prove  that 
the  two  pistons  in  the  diagram  are  in  equilibrium  of  this  kind.  For 
their  common  centre  of  gravity  is  at  a  point  which  divides  the  line  of 
their  respective  centres  of  gravity  in  the  proportion  of  their  weights. 
Now  if  they  move,  if  this  is  possible,  their  paths  will  be  related  to  each 
other  as  their  reciprocal  weights,  as  we  have  shown.  Now  if  their  com 
mon  centre  of  gravity  is  taken  in  this  second  situation,  it  will  be  found  in 
exactly  the  same  position  as  previously.  Therefore  the  two  pistons, 
considered  as  one  and  the  same  body,  move  in  such  a  way  that  their 
centre  of  gravity  does  not  descend,  which  is  contrary  to  the  principle. 
Therefore  they  are  in  equilibrium.  Q.E.D." 

The  reason  for  the  equilibrium  in  all  Pascal's  examples  lies  in  the 
fact  that  "  the  material  which  is  extended  over  the  base  of  the  vessels, 
from  one  opening  to  the  other,  is  liquid.  "  In  mechanics  this  property 
which  belongs  to  incompressible  fluids  of  wholly  transmitting  a  pressure 
has  been  called  Pascal's  Principle, 

We  shall  not  detain  ourselves  further  with  Pascal's  Treatise — which 


DESCARTES   AND   PASCAL  171 

has  become  classical — nor  with  the  suggestive  forms  that  he  gave 
the  hydrostatic  paradox.  Stevin  had  anticipated  some  of  these  ideas. 
Pascal's  contribution  to  physics  was  in  marked  contrast  with  Des 
cartes9  fragile  conceptions  in  the  same  field.  However,  it  had  its  con 
temporary  opponents — thus  Pascal  was  obliged  to  contend  with  Aristo 
telians  like  Father  NoeL  At  first  Pascal  took  various  verbal  precautions 
when  he  argued  the  futility  of  the  horror  of  the  vacuum.  But  eventually 
he  became  forthright  in  his  conviction  that  this  scholastic  prejudice  was 
absurd.  "  Let  all  the  disciples  of  Aristotle  gather  together  all  the 
strength  in  the  writings  of  their  master  and  his  commentators  in  order, 
if  they  can,  to  make  these  things  reasonable  by  means  of  the  horror  of 
the  vacuum.  Except  that  they  know  that  experiments  are  the  true 
masters  that  must  be  followed  in  physics.  And  that  what  has  been 
accomplished  in  the  mountains  reverses  the  common  belief  of  the  world 
that  Nature  abhors  a  vacuum  ;  it  has  also  established  the  knowledge 
— which  will  never  die — that  Nature  has  no  horror  of  a  vacuum,  and 
that  the  heaviness  of  the  mass  of  air  is  the  true  cause  of  all  the  effects 
which  have  previously  been  attributed  to  this  imaginary  cause.  " 


CHAPTER   FIVE 


THE  LAWS  OF  IMPACT 

(WALLIS,  WREN,  HUYGHENS,  MARIOTTE) 

THE  MECHANICS  OF  HUYGHENS 

(1629-1697) 


1.  THE  MECHANICS  OF  WALLIS  (1616-1703). 

In  1668  the  Royal  Society  of  London  took  the  initiative  of  choosing 
for  discussion  the  subject  of  the  laws  of  impacting  bodies.  Wallis 
(November  26th,  1668)  discussed  the  impact  of  inelastic  bodies,  while 
Wren  (December  17th,  1668)  and  Huyghens  (January  4th,  1669)  dis 
cussed  the  impact  of  elastic  bodies. 

Wallis'  memoir  should  be  set  against  the  background  of  his  other 
work  in  mechanics,  which  was  the  subject  of  a  treatise  Mechanic^  sive 
de  Motu  (London,  1669-1671).  In  this  treatise  Wallis  considered  the 
vis  matrix  and  the  resistantia,  which  were  opposed  to  each  other  in 
every  machine.  He  hesitated  between  the  concept  of  memento  in 
Galileo's  sense,  which  could  be  expressed  by  the  product  pvs  of  the 
weight  and  the  vertical  component  of  the  velocity,  and  the  notion  of 
moment,  or  the  product  ph  of  the  weight  and  the  height  of  fall.  In  the 
first  Chapter,  Wallis  even  went  as  far  as  to  consider  a  mixed  solution 
by  the  product  pvs  as  a  measure  of  the  momentum  of  the  motive  force 
and  the  product  ph  as  a  measure  of  the  impedimentum  of  the  resistance. 
Very  fortunately  he  did  not  adhere  to  this  peculiar  choice  and  declared 
himself  in  favour  of  the  product  ph  in  the  second  Chapter  of  his  Treatise, 
which  was  concerned  with  the  fall  of  bodies.  He  went  further  and 
generalised  the  principle  that  Descartes  had  laid  at  the  foundation  of 
statics  by  extending  it  to  forces  other  than  the  heaviness. 

"  In  an  absolutely  general  way,  the  progress  effected  by  a  motive 
force  is  measured  by  the  movement  effected  in  the  direction  of  this 
force,  the  recoil  by  the  movement  in  the  opposite  direction.  .  .  .  The 
progress  or  recoil  effected  under  the  action  of  any  force  is  obtained  by 


WALLIS,   WREN,   HUYGHENS,   MARIOTTE  173 

taking  the  products  of  the  forces  by  the  lengths  of  the  progress  or  recoil 
reckoned  in  the  line  of  direction  of  the  force.  " 

Here  Wallis  supposes  the  displacements  to  be  finite  and  rectilinear 
and  the  forces  to  be  constant  in  magnitude  and  direction.  But  he 
assumes  that  a  curvilinear  trajectory  can  be  represented  as  a  limit  of  its 
tangents.  It  is  said  that  Wallis  also  exploited  the  notion  of  work  in 
all  its  generality,  in  so  far  as  displacements  due  to  a  constant  force  were 
concerned. 

Without  delaying  ourselves  over-much  with  this  treatise,  we  shall 
discuss  Wallis's  treatment  of  the  laws  of  impact.  Wallis  called  a  body 
perfectly  hard  if  it  did  not  yield  in  any  way  in  impact.  This  is  a  category 
that  must  be  distinguished  from  soft  and  elastic. 

"  A  soft  body  is  one  that  yields  at  impact  in  such  a  way  as  to  lose 
its  original  shape,  like  clay,  wax,  lead.  .  .  .  For  these  bodies  part  of  the 
force  is  used  to  deform  them — the  whole  force  is  not  expended  in  the 
obstacle.  It  is  necessary  to  take  account  of  this  part.  " 

Like  Jouguet,1  we  remark  that  from  the  point  of  view  of  energy, 
Wallis  had  good  reason  to  make  this  distinction.  According  to  whether 
the  internal  energy  of  a  body  depends  on  its  deformation  or  not,  the 
living  force  lost  in  the  impact  is  not  equivalent,  or  is  equivalent,  to  the 
heating  of  the  body.  But  from  the  point  of  view  of  the  quantity  of 
motion,  which  was  the  one  that  Wallis  took,  this  energetic  distinction 
is  irrelevant.  Mariotte  abandoned  it,  and  was  followed  by  others. 

Wallis  called  a  body  elastic  if  it  yielded  in  impact,  but  then  sponta 
neously  regained  its  original  shape,  like  a  steel  spring. 

Finally,  he  defined  the  direct  impact  of  two  bodies  in  the  way  that 
is  now  accepted. 

We  now  come  to  Wallis's  proposition  relating  to  soft  impact  (Chapter 
XI,  Proposition  II). 

"  If  a  body  in  motion  collides  with  a  body  at  rest,  and  the  latter  is 
such  that  it  is  not  moving  nor  prevented  from  moving  by  any  external 
cause,  after  the  impact  the  two  bodies  will  go  together  with  a  velocity 
which  is  given  by  the  following  calculation. 

"  Divide  the  momentum  furnished  by  the  product  of  the  weight  and 
the  velocity  of  the  body  which  is  moving  by  the  weight  of  the  two 
bodies  taken  together.  You  will  have  the  velocity  after  the  impact. 

"  Indeed,  let  a  body  A  be  in  motion  along  the  line  AA  that  passes 
through  its  centre  of  gravity  and  through  that  of  the  body  B  which  is 
at  rest.  Let  p  and  v  be  the  weight  and  the  velocity  of  A.  The  impelling 
force  (vis  impellens)  will  be  pv.  Let  p'  be  the  weight  of  the  body  B 

1  L.  M.9  Vol.  I,  p.  123. 


174  THE   FORMATION   OF   CLASSICAL  MECHANICS 

whose  velocity  is  nothing.  The  weight  of  the  two  bodies  is  p  +  p'. 
After  the  impact  the  two  bodies  will  move  with  the  same  velocity. 
Indeed,  JB  cannot  go  more  slowly  than  A9  since  A  follows  it.  Neither 
can  it  go  more  quickly,  for  it  is  supposed  that  there  is  no  other  cause  of 
motion  than  that  which  arises  from  the  impulsion  of  A.  (If  there  is 
another  force,  like  the  elastic  force,  which  can  impell  B  more  quickly, 
the  problem  is  of  another  order,  to  which  we  shall  return.)  Therefore 

the  weight  p  +  p'  is  moved  by  a  force  pv  and  its  velocity  is  — — — -.  " 

P  +  P 

This  demonstration  is  based  on  the  conservation  of  the  total  quantity 
of  motion  of  the  system.  It  has  already  been  remarked  that  Wallis  did 
not  distinguish  between  the  weight  and  the  mass. 


Fig.  61 

Wallis  generalised  this  argument  to  the  situation  in  which  J5  is  in 
motion  with  a  smaller  velocity  than  that  of  A,  but  in  the  same  direction. 
If  vf  is  the  velocity  of  JS,  the  common  velocity  of  the  two  bodies  A  and 
B  after  the  impact  is 

pv  +  p'v' 

This  is  always  obtained  by  dividing  the  sum  of  the  moments  by  the 
sum  of  the  weights. 

If  the  velocity  of  B  is  in  the  opposite  direction  to  the  velocity  v  of 
A,  and  is  denoted  by  — i/,  the  common  velocity  of  the  two  bodies  A 
and  B  after  the  impact  is 

pv  —  p'vf 

P+P'   ' 

It  is  seen  that  Wallis,  unlike  Descartes,  was  careful  to  take  account  of 
the  signs  of  the  quantities  of  motion.  That  is  why  he  arrived  at  rules 
which  were,  apart  from  the  confusion  of  the  weight  and  the  mass,  correct. 
Finally,  Wallis  remarked  that  "  the  magnitude  of  the  impact  is  equal  to 
twice  the  decrease  that  is  experienced  by  the  greatest  moment  in  direct 
impact.  " 

Indeed,  "  Consider  the  body  which  has  the  greatest  moment  as 
hitting,  and  the  other  body  as  being  hit.  The  body  which  is  hit  receives 


WALLIS,   WREN,   HUYGHENS,   MARIOTTE  175 

as  much  moment  as  is  lost  by  the  body  which  hits  it.  These  moments 
that  are  gained  or  lost  are  both  the  restdt  of  the  impact.  The  impact 
is  therefore  equal  to  their  sum,  that  is,  to  twice  the  decrease  experienced 
by  the  greater  moment.  " 

Wallis  was  also  concerned  with  elastic  impact  (in  Chapter  XIII  of 
his  Treatise).  He  related  this  to  the  theory  of  soft  impact. 

He  introduced  an  elastic  force  (vis  elastica)  whose  nature  he  did  not 
specify,  confining  himself  to  an  appeal  to  experimental  facts.  He 
stated  the  following  proposition. 

"  If  a  body  hits  an  obstacle  directly,  and  if  the  two  bodies — or  only 
one  of  them — are  elastic,  the  first  body  will  rebound  with  a  velocity 
equal  to  that  which  it  had  before  the  impact,  and  will  follow  the  same 
direction.  " 

Indeed,  "  if  the  elasticity  were  nothing,  the  body  A  would  come  to 
rest.  " — (This  conclusion  is  obtained  by  applying  the  theory  of  soft 
impact  to  an  immovable  obstacle  B.  Moreover,  we  add  that  Wallis 
extended  this  result  to  an  obstacle  whose  force  of  resistance  was  limited 
by  comparing  this  force  with  a  moment  greater  than  that  of  the  body  A.) 
Wallis  continues — 

44  Therefore  all  motion  remaining  after  the  impact  is  the  result  of  the 
vis  elastica.  Now  this  is  always  equal  to  the  force  of  the  impact.  .  .  . 
Indeed,  the  elasticity  does  not  resist  as  a  simple  impedimentum,  but 
rather  as  a  contrary  force  acting  by  reaction  and  with  the  same  energy 
that  the  compression  requires.  Now  what  the  elasticity  suffers  during 
the  compression  is  equal  to  the  impact.  The  restitutive  force  is  there 
fore  equal  to  the  impact.  .  .  .  Now,  in  particular,  since  the  body  A 
has  weight  p  and  velocity  t;,  the  magnitude  of  the  impact  is  2pv.  This 
is  also  that  of  the  elastic  force.  Since  it  is  developed  equally  in  the  two 
parts  half  of  this  force,  that  is  pv,  acts  on  the  obstacle  and  is  dissipated 
there,  while  the  other  half  repels  the  body  A  with  velocity  v.  " 

Wallis  also  treats,  for  example,  the  elastic  collision  of  two  equal 
bodies  which  have  equal  and  opposite  velocities  and — borrowing  this 
from  Wren — the  collision  of  unequal  bodies  with  velocities  inversely 
proportional  to  their  weights.  Each  body  rebounds  with  the  velocity 
that  it  has  before  the  impact.  These  demonstrations  are  analogous 
to  the  preceding  one  and  we  shall  not  describe  them  here. 

2.  WREN  (1632-1723)  AND  THE  LAWS  OF  ELASTIC  IMPACT. 

We  shall  describe  the  paper  presented  by  Wren  at  the  meeting  in 
1668  and  which  is  included  in  the  Philosophical  Transactions  of  1669. 
Wren  starts  from  the  concept  of  proper  velocity  which,  for  any  body, 


176  THE   FORMATION   OF   CLASSICAL   MECHANICS 

is  inversely  proportional  to  the  weight.  The  impact  of  two  bodies  R 
and  S  which  travel  with  their  proper  velocities  results  in  the  conserva 
tion  of  these  velocities.  If  the  velocities  differ  from  their  proper  values, 
the  bodies  R  and  S  "  are  brought  back  to  equilibrium  by  the  impact.  " 
This  is  to  say  that  if,  before  the  impact,  the  velocity  of  R  is  greater  than 
its  proper  velocity  by  a  certain  amount  and  that  of  S  is  less  than  its 
proper  velocity  by  the  same  amount,  as  a  result  of  the  impact  this 
amount  is  added  to  the  proper  velocity  of  S  and  subtracted  from  that 

of  JR. 

Wren  seems  to  have  regarded  the  impact  of  two  bodies  with  their 
proper  velocities  as  equivalent  to  a  balance  oscillating  about  its  centre 
of  gravity.  Wren  expressed  this  analogy  in  the  diagrams  that  he  used 
to  represent  the  effect  of  the  impact.  Strictly  speaking,  he  did  not 
justify  his  results  in  a  logical  and  satisfactory  way,  but  he  had  the  merit 
of  making  experiments  and  of  embodying  his  conclusions  in  a  clear  and 
precise  law. 


3.  HUYGHENS  (1629-1697)  AND  THE  LAWS  OF  IMPACT. 

Following  Wren's  example,  Huyghens  confined  himself  to  elastic 
impact.  His  researches  were  collected  in  a  posthumous  volume  De 
Motu  corporum  ex  percussione  (1700). 

Huyghens'  investigation  was  based  on  the  following  three  hypo 
theses. 

1)  The  first  is  the  principle  of  inertia.     "  Any  body  in  motion  tends 
to  move  in  a  straight  line  with  the  same  velocity  as  long  as  it  does  not 
meet  an  obstacle.  " 

2)  The  second  is  the  following  principle.     Two  equal  bodies  which 
are  in  direct  impact  with  each  other  and  have  equal  and  opposite  velocities 
before  the  impact,  rebound  with  velocities  that  are,  apart  from  sign,  the 
same. 

3)  The  third  hypothesis  asserts  the  relativity  of  motion.     Huyghens 
shows  himself  to  be  a  Cartesian  in  this  matter. 

"  The  expressions  4  motion  of  bodies  '  and  '  equal  or  unequal  velo 
cities  '  should  be  understood  relatively  to  other  bodies  that  are  consider 
ed  as  at  rest,  although  it  may  be  that  the  second  and  the  first  both  parti 
cipate  in  a  common  motion.  And  when  two  bodies  collide,  even  if  both 
are  subject  to  a  uniform  motion  as  well,  to  an  observer  who  has  this 
common  motion  they  will  repel  each  other  just  as  if  this  parasitica1 
motion  did  not  exist. 


WALLIS,   WREN,   HUYGHENS,    MARIOTTE  177 

"  Thus  let  an  experimenter  be  carried  by  a  ship  in  uniform,  motion 
and  let  him  make  two  equal  spheres,  that  have  equal  and  opposite 
velocities  with  respect  to  him  and  the  ship,  collide.  We  say  [hypo 
thesis  2]  that  the  two  bodies  will  rebound  with  velocities  that  are  equal 
with  respect  to  the  ship,  just  as  if  the  impact  were  produced  in  a  ship  at 
rest  or  on  terra  firma.  " 

Huyghens  appealed  to  this  relativity  in  order  to  justify  the  following 
proposition. 

"  Proposition  /.  —  If  a  body  is  at  rest  and  an  equal  body  collides  with 
it,  after  the  impact  the  second  body  will  be  at  rest  and  the  first  will  have 
acquired  the  velocity  that  the  other  had  before  the  impact. 

"  Imagine  that  a  ship  is  carried  alongside  the  bank  by  the  current 
of  a  river  and  that  it  is  so  close  to  the  edge  that  a  passenger  on  the  ship 
can  hold  the  hands  of  an  assistant  on  the  bank.  In  his  two  hands,  A 
and  B,  the  passenger  holds  two  equal  bodies  E  and  F  which  are  hung 
from  threads.  Let  the  distance  EF  be  divided  into  two  equal  parts  by 
the  point  G.  By  displacing  his  two  hands  equally  towards  each  other, 
the  passenger  will  make  the  two  spheres  E  and  F  collide  with  equal 
velocities  with  respect  to  himself  and  the  ship  (hypotheses  2  and  3). 
But  during  this  time  the  ship  is  supposed  to  be  carried  to  the  right  with 
a  velocity  GE  equal  to  that  with  which  the  right  hand  of  the  passenger 
is  moved  towards  the  left. 

"  Consequently  the  right  hand,  A,  is  motionless  with  respect  to  the 
bank  and  the  assistant  who  is  placed  there,  while  the  passenger's  left 
hand,  B,  is  displaced  with  a  velocity  EF — twice  of  GE  or  FG — with 
respect  to  the  assistant.  Suppose  that  the  assistant  placed  on  the  bank 
grasps  the  passenger's  hand  A^  as  well  as  the  end  of  the  thread  which 
supports  the  globe  jB,  with  his  own  hand  C.  Also,  that  with  his  hand  D 
he  grasps  the  passenger's  hand  B,  which  is  the  one  that  holds  the  thread 
from  which  the  sphere  F  hangs.  It  is  seen  that  when  the  passenger 
makes  the  spheres  E  and  F  meet  each  other  with  velocities  that  are 
equal  with  respect  to  himself  and  the  ship,  at  the  same  time  his  assistant 
makes  the  sphere  E — motionless  with  respect  to  himself  and  the  bank — 
collide  with  the  sphere  jF  whose  velocity  is  FE.  And  it  is  certain  that, 
if  the  passenger  displaces  the  spheres  in  the  way  that  has  been  described, 
there  is  nothing  to  prevent  his  assistant  on  the  bank  from  seizing 
his  hands  and  the  ends  of  the  threads,  provided  only  that  he  accompanies 
the  motion  and  does  not  oppose  any  hindrance  to  them.  In  the  same 
way,  when  the  assistant  on  the  bank  is  directing  the  sphere  F  against 
the  motionless  sphere  E,  there  is  no  obstacle  to  the  passenger  grasping 
his  hands,  even  though  the  hands  A  and  C  are  at  rest  with  respect  to 
the  bank  while  the  hands  D  and  JB  move  with  the  same  velocity  EF. 

12 


178 


THE   FORMATION   OF   CLASSICAL  MECHANICS 


"  As  we  have  seen,  the  spheres  E  and  F  rebound  after  the  impact 
with  velocities  that  are  equal  with  respect  to  the  passenger  and  the 
ship — that  is,  the  sphere  E  with  the  velocity  GE  and  the  sphere  F  with 
the  velocity  GF.  During  this  time  the  ship  moves  towards  the  right 


passenger's  hand 
t's  hand 


O 


F 


O 


Fig.  62 


with  the  velocity  GE  or  FG.  Therefore,  with  respect  to  the  bank  and 
the  assistant  on  it,  the  sphere  F  remains  motionless  after  the  impact 
and  the  sphere  E  moves  to  the  left  with  a  velocity  twice  GJ5 — that  is, 
with  the  velocity  FE  with  which  F  has  hit  E.  We  therefore  show  that, 
to  an  observer  on  the  earth,  when  a  motionless  body  is  hit  by  an  equal 
one,  after  the  impact  the  second  one  loses  all  its  motion  which  is,  on  the 
other  hand,  completely  taken  over  by  the  other.  " 

With  the  help  of  this  remarkable  artifice  Huyghens  treats  all 
instances  of  the  impact  of  two  equal  bodies  by  starting  from  the  symme 
trical  case,  whose  solution  he  assumes  a  priori.  Thus  he  shows  that 
two  equal  bodies  that  have  unequal  velocities  exchange  these  velocities 
in  a  direct  impact. 

He  then  passes  to  a  consideration  of  unequal  bodies  and  establishes 
the  conservation  of  the  relative  velocity  in  the  impact  of  two  elastic  bodies. 
It  is  of  some  interest  to  remark  here  that  Huyghens  showed,  by  examples, 
that  the  quantity  of  motion  was  not  always  conserved.  In  this  context 
he  was  concerned  with  the  quantity  of  motion  m|  v|,  in  Descartes'  sense. 

Finally,  he  demonstrates  Wren's  rule  on  the  conservation  of  proper 
velocities. 

"  Proposition  VIII.  —  If  two  bodies  moving  in  opposite  directions 
and  with  velocities  inversely  proportional  to  their  magnitudes  collide  with 
each  other,  each  one  rebounds  with  the  velocity  that  it  had  before  the  impact. 


WALLIS,   WREN,   HUYGHENS,   MARIOTTE 


179 


"  Let  two  bodies  A  and  JB  collide  with  each  other.  (A  >  B). 
Suppose  that  the  velocity  BC  of  the  body  B  is  to  the  velocity  AC  of  the 
body  A  as  the  magnitude  A  is  to  the  magnitude  B.  We  wish  to  show 
that  after  the  impact,  A  will  be  reflected  with  a  velocity  CA  and  B 
with  a  velocity  CD.  If  the  former  is  true  for  A9  the  latter  is  true  for  B 
(conservation  of  relative  velocity).  Suppose  that  A  is  reflected  with 


Fig.  63 


a  velocity  CD  <  CA.  Then  B  will  rebound  with  a  velocity  CE  >  CB 
and  DE  =  AB.  Imagine  that  A  has  acquired  its  original  velocity  AC 
by  falling  from  the  height  HA  and  that  its  vertical  motion  has  then  been 
changed  into  the  horizontal  motion  with  velocity  A  C.  In  the  same  way, 
suppose  that  B  has  acquired  its  velocity  by  falling  from  the  height  KB. 
These  heights  are  in  the  square  ratio  of  the  velocities.  That  is, 

HA       fCA^ 2 

— --  =  J  — -  j  .     Then  suppose  that  after  the  impact  the  bodies  A  and  B 

KB       \CB  J 

change  their  horizontal  motions — whose  velocities  are  CD  and  CE — 
into  vertical  upwards  motions  and  thus  arrive  at  the  points  L  and  M 

,    ,        AL        /CD\2 
such  that  m=(m). 

"  When  the  centre  of  gravity  of  A  is  at  H  and  that  of  B  at  jK, 
their  common  centre  of  gravity  is  at  Q.  After  the  impact  this 
centre  of  gravity  is  at  the  point  N.  Now  it  can  be  shown  that  IV  is 
above  Q.  "  * 

In  this  matter  Huyghens  invokes  a  principle  which  we  find  developed 
in  some  detail  in  the  Horologium  oscillatorium^  and  reduces,  in  fact, 
to  the  principle  of  the  conservation  of  living  forces. 

1  This  is  a  question  of  pure  geometry. 

2  See  below,  p.  187. 


180  THE    FORMATION   OF   CLASSICAL   MECHANICS 

66  It  is  a  well-established  principle  of  Mechanics  that,  in  the 
motion  of  several  bodies  under  the  influence  of  their  centre  of 
gravity  alone,  the  common  centre  of  gravity  of  these  bodies  cannot 
be  raised.  " 

If  this  principle  is  assumed,  the  supposition  which  has  been  made 
about  the  velocity  with  which  A  rebounds  (CD  <  CA)  implies  a  con 
tradiction.  Huyghens  dismisses  the  hypothesis  that  A  is  reflected  with 
a  velocity  CD  >  CA  in  an  analogous  way.  Therefore  A  rebounds  with 
the  velocity  CA  and  B  with  the  velocity  CB.  Q.  E.  D. 

Huyghens  related  all  cases  of  the  direct  elastic  impact  of  two  unequal 
bodies  to  the  preceding  situation  by  using  the  artifice  of  the  moving 
ship  on  every  occasion. 

We  now  know  that  by  invoking  the  relativity  of  impact  phenomena, 
Huyghens  carried  the  discussion  into  a  privileged  field.  It  follows  from 
the  rule  of  the  composition  of  velocities  that  a  percussion  remains  the 
same  when  a  "  fixed  "  system  of  reference  is  replaced  by  a  "  moving  " 
one,  from  the  moment  when  the  relative  motion  of  the  two  systems 
becomes  continuous.  With  this  restriction  alone,  the  relative  motion 
of  the  two  systems  can  be  accelerated  in  any  way.  Huyghens,  however, 
restricted  himself  to  a  uniform  and  rectilinear  relative  motion — namely, 
that  of  the  ship  with  respect  to  the  river-bank. 

We  add  that  in  using  the  principle  of  inertia,  on  the  other  hand, 
Huyghens  confined  himself  to  an  infinite  number  of  reference  points, 
to  day  termed  absolutes.  In  fact,  the  principle  of  inertia  is  irrelevant 
to  impact  phenomena  because  of  their  instantaneous  character. 

In  all  his  writings,  Huyghens  took  care  to  explain  his  hypotheses 
clearly  and  to  deduce  his  propositions  from  them  logically.  His  style 
is  similar  to  that  of  Archimedes.  By  this  means  a  perfect  clarity  is 
achieved  at  the  price  of  some  tedium.  However,  the  rigour  of  this  work 
is  sometimes  only  an  apparent  one. 

Jouguet  has  come  to  the  following  conclusion  after  an  exhaustive 
study  of  the  interdependence  of  certain  of  Huyghens'  hypotheses.  1 
It  was  sufficient  for  Huyghens  to  assume  the  conservation  of  the  total 
living  force  of  the  system  in  every  system  of  reference,  or  its  conservation 
in  two  arbitrary  systems  in  continuous  (and  not  zero)  relative  motion 
with  respect  to  each  other.  Such  a  hypothesis  is  itself  equivalent  to 
the  twin  hypothesis  of  the  conservation  of  living  force  in  one  arbitrary 
system  of  reference  and  the  simultaneous  conservation  of  the  total 
quantity  of  motion  in  direction  and  sign  as  Wallis  intended — not  in 
Descartes'  sense  of  absolute  value. 

1  L.  M.,  Vol.  I,  p.  151. 


WALLIS,   WREN,   HUYGHENS,   MARIOTTE  181 

4.    THE   PLAN   OF   HUYGHENS'  FUNDAMENTAL  TREATISE. 

We  now  come  to  Huyghens*  major  work  in  dynamics — the  treatise 
Horologium  oscillatorium  sive  de  motu  pendulorum  ad  horologia  aptato 
demonstrationes  geometricae  (Paris,  1673). 1 

This  work  consists  of  five  parts —  a  description  of  the  clock;  on  the 
fall  and  motion  of  bodies  on  a  cycloid ;  the  evolution  and  dimensions  of 
curved  lines ;  on  the  centre  of  oscillation  or  agitation ;  and  finally,  on 
the  construction  of  a  new  clock  with  a  circular  pendulum,  and  theorems 
on  the  centrifugal  force. 

Huyghens  had  constructed  cycloidal  clocks  at  The  Hague  since  1657. 
He  planned  this  treatise  over  a  long  period  of  time  and  only  completed 
it  in  1669.  In  1665  he  was  invited  hy  Colbert  to  visit  Paris  and  to  work 
at  the  Academy  of  Sciences,  where  he  obtained  a  royal  monopoly  for 
the  reproduction  of  his  clocks.  As  early  as  1667  there  is  mention  of 
"  three  clocks  made  at  Paris,  at  the  expense  of  the  King,  to  be  used  on 
the  voyage  to  Madagascar.  "  Huyghens  had  already  tried  out  his 
marine  clock  aboard  an  English  vessel  in  1664.  Aware  of  the  defect  of 
the  isochronism  of  the  finite  oscillations  of  a  circular  pendulum, 
Huyghens  strove  to  find  a  pendulum  which  might  be  theoretically 
isochronous  for  all  amplitudes.  "  It  is  the  oscillations  of  marine  clocks 
that  most  noticeably  become  unequal,  because  of  the  ship's  continual 
shaking.  So  that  it  is  necessary  to  take  care  that  oscillations  of  large 
and  small  amplitudes  should  be  isochronous.  "  Huyghens  also  made 
astronomical  clocks,  both  at  Leyden  and  Paris,  that  were  correct  to  one 
second  a  day. 

"  We  have, "  he  wrote, "  regarded  the  cyloid  as  the  cause  of  this  pro 
perty  of  isochronism  that  we  have  found,  without  having  the  least 
understanding  of  anything  except  that  it  is  consistent  with  the  rules 
of  the  craft.  "  In  theory,  "  it  has  been  necessary  to  corroborate  and 
to  extend  the  doctrine  of  the  great  Galileo  on  the  fall  of  bodies.  The 
most  desired  result  and,  so  to  speak,  the  greatest,  is  precisely  this  pro 
perty  of  the  cyloid  that  we  have  discovered.  ...  To  be  able  to  relate 
this  property  to  the  use  of  pendulums,  we  have  had  to  study  a  new 
theory  of  curved  lines  that  produce  others  in  their  own  evolution.  " 
The  question  here  is  that  of  the  theory  of  developable  curves,  and 
Huyghens  established  that  the  development  of  a  cycloid  is  an  equal 
cycloid. 

Huyghens  describes  his  "  automatic  "  at  great  length.  In  this  clock 
the  motion  of  the  pendulum  is  determined  by  pulleys  that  are  actuated 

1  The  complete  Works  of  Christiaan  Huyghens,  published  by  the  Dutch  Society  of 
the  Sciences,  The  Hague,  1934,  Vol.  XVII. 


182  THE   FOBMATION   OF   CLASSICAL  MECHANICS 

by  the  driving  weights —  "  the  oscillations  of  the  pendulum  impose  the 
law  and  the  rule  of  motion  on  the  whole  clock.  "  He  also  evolved  a 
new  and  ingenious  suspension  which  assured  the  continuity  of  the 
clock's  motion  when  the  driving  weights  were  removed  and  replaced. 


Fig.  64 


We  return  to  the  question  of  the  cycloidal  pendulum.  The  pendu 
lum  oscillates  between  two  thin  plates  whose  function  is  to  assure  the 
constancy  of  the  period  in  spite  of  variations  in  the  amplitude.  These 
plates,  Km  and  Ki9  are  cut  from  two  half- cycloids,  KM  and  KI. 
The  pendulum  KMP  has  a  length  equal  to  twice  the  diameter  of  the 
generating  circle.  Huyghens  wrote,  "I  do  not  know  whether  any 
other  line  has  this  remarkable  property,  namely  that  of  describing 
itself  in  its  evolution  " — in  other  words,  of  being  identical  with  its 
development.  This  mathematical  problem  was  to  be  taken  up  by 
Euler. 


5.  HUYGHENS  AND  THE  FALL  OF  BODIES. 

Though,  as  we  have  remarked  in  connection  with  the  laws  of  impact, 
Huyghens  was  a  Cartesian  to  the  extent  that  he  used  the  relativity 
of  motion,  he  was  much  more  a  direct  successor  of  Galileo  and  Torri- 
celli,  and  provided  a  link  between  them  and  Newton.  To  use  his 
words,  he  "  corroborated  and  extended  Galileo  "  in  the  matter  of  the 
fall  of  bodies. 

He  clearly  stated  the  principle  of  inertia  and  the  principle  of  the 
composition  of  motions,  and  applied  these  principles  to  the  fall  of 
bodies  and  to  rectilinear  uniform  motion  in  any  direction.  "  Each  of 
these  motions  can  be  considered  separately.  One  does  not  disturb 


WALLIS,   WREN,   HUYGHENS,   MARIOTTE  183 

the  other.  "  He  accepted  Galileo's  laws  on  the  rectilinear  fall  of  bodies 
and  improved  the  associated  demonstrations.  For  example,  he 
established  the  following  proposition. 

"  Proposition  I.  —  In  equal  times  the  increases  of  the  velocity  of  a 
body  which  starts  from  rest  and  falls  vertically  are  always 
equal,    and   the   distances   travelled   in   equal   times  form  ,  ^ 

a  series  in  which  successive  differences  are  constant.  M     n  ,  „ 

"  Suppose  that  a  body,  starting  from  rest  at  A,  falls 
through  the  distance  AB  in  the  first  time  and  has  acquir- 
ed  a  velocity  at  JB  which  would  allow  it  to  travel  the 
distance  BD  during  the  second  time.  \£ 

"  We  know  then  that  the  distance  travelled  in  the 
second  time  will  be  greater  than  BD,  since  the  distance 
BD  would  be  travelled  even  if  all  the  action  of  the  weight 
ceased  at  B.  In  fact,  the  body  will  be  animated  by  a 
compound  motion  consisting  of  the  uniform  motion 
which  would  allow  it  to  travel  the  distance  BD  and  the 
motion  of  the  fall  of  bodies,  by  means  of  which  it  must 
necessarily  fall  a  distance  AB.  Therefore  at  the  end 
of  the  second  time,  the  body  will  arrive  at  the  point  E 
which  is  obtained  by  adding  to  BD  a  length  DE  which 
is  equal  to  AB.  ..." 

Huyghens  showed  in  the  same  way  that,  at  the  end 
of  the  third  time,  the  body  will  arrive  at  G  —  a  point 
such  that  EF—2BD  and  FG=AB.  The  same  procedure 
was  repeated. 

Thus  Huyghens  arrived  at  the  following  proposition.  Fl&-  65 

"  Proposition  II.  —  The  distance  that  a  body  starting 
from  rest  travels  in  a  certain  time  is  half  the  distance  it  would  travel  in 
uniform  motion  with  the  velocity  acquired,  in  falling,   at  the  end  of  the 
time  considered.  " 

To  show  this,  Huyghens  considers  the  distances  AB,  BE,  EG  and 
GK  travelled  in  the  first  four  intervals  of  time.  He  doubles  the  value 
of  these  times  so  that  the  body  travels  along  AE  in  the  first  instant 
and  along  EK  in  the  second.  Necessarily 


„ 


BE  =  AD 
BE       EK  AE    ~  AB  ~~~  AB 


Now  KE=2AB  +  5BD       and 

Therefore  KE—EA  =  4BD 


184  THE   FORMATION    OF    CLASSICAL   MECHANICS 

4BD       AD—AB       BD 
and  consequently      __  =  ___=—. 

Therefore  AE  =  4>AB     and    BD  =  2AB.     Q.  E.  D. 

These  demonstrations  have  been  quoted  because  they  differ  from 
those  of  Galileo.  In  particular,  they  make  use  of  a  composition  of 
the  velocity  acquired  and  the  new  fall  of  the  body  at  each  instant. 
With  a  little  good-will  it  is  possible  to  regard  them  as  the  expression 
of  the  ideas  of  the  Schoolmen  of  the  XlVth  Century  in  a  more  sophis 
ticated  mathematical  language.  Thus  Buridan,  in  particular,  believed 
that  heaviness  continually  caused  a  new  impetus  to  the  one  that  was 
already  present. 

Huyghens  then  sets  out  to  establish  a  hypothesis  "  that  Galileo 
asked  should  be  granted  to  him  as  obvious.  "  Thus  Salviati,  in  the 
Discorsi,  had  been  obliged  to  take  the  following  principle  as  a  postulate.1 

"  The  velocities  acquired  by  a  body  in  falling  on  differently  inclined 
planes  are  equal  when  the  heights  of  the  planes  are  so.  " 


B 


D 


Fig.  66 


"...  Let  there  be  two  inclined  planes  whose  sections  by  a  vertical 
plane  are  AB  and  CB.  Their  heights,  AE  and  CD,  are  equal.  I  maintain 
that  in  these  two  circumstances  the  velocity  acquired  at  B  is  the  same. 
Indeed,  if  in  falling  along  CB  the  body  acquires  a  velocity  that  is  smaller 
than  in  falling  along  AB,  this  velocity  will  be  equal  to  that  which 
would  be  acquired  in  some  descent  FB  <  AB.  But  along  CB  the  body 
acquires  a  velocity  that  allows  it  to  rise  again  along  the  whole  length 
of  BC.  [This  in  virtue  of  proposition  IV,  which  we  have  not  quoted 
and  which  demonstrates  this  fact  for  a  rectilinear  rise.]  Therefore 
it  will  acquire,  along  FB,  a  velocity  which  can  make  it  rise  again  along 
BC — this  can  be  achieved  by  reflection  at  an  oblique  surface.  It 
will  therefore  rise  as  far  as  C,  or  to  a  height  greater  than  that  from 
which  it  fell,  which  is  absurd.  " 

1  See  above,  p.  135. 


WALLIS,   WREN,   HUYGHENS,   MARJOTTE  185 

In  the  same  way  it  is  shown  that  in  descending  along  AB  the  body 
cannot  acquire  a  smaller  velocity  than  in  falling  along  CB.  This 
establishes  the  proposition. 

Huyghens  shows  that  the  durations  of  fall  have  the  same  relation 
to  each  other  as  the  lengths  of  the  planes.  He  also  shows  that  when 
the  body  falls  in  a  continuous  motion  from  a  given  height  along  any 
number  of  differently  inclined  planes,  it  always  acquires  the  same 
velocity  as  that  obtaining  at  the  end  of  a  vertical  fall  from  the  same 
height.  Conversely,  in  rising  again  along  a  trajectory  formed  of 
contiguous  and  differently  inclined  planes,  the  body  will  achieve  its 
original  height  (Proposition  IX).  A  passage  to  the  limit  then  allows 
the  question  of  the  motion  of  a  body  on  a  curve  contained  in  a  vertical 
plane  to  be  considered. 

6.  THE  ISOCHRONISM  OF  THE  CYCLOIDAL  PENDULUM. 

Huyghens  arrived  at  a  proof  of  the  isochronism  of  a  cycloidal 
pendulum  by  means  of  an  argument  in  infinitesimal  geometry.  Ad 
mirable  though  this  was,  it  required  no  less  than  a  dozen  propositions, 
and  we  cannot  reproduce  it  here.1  The  principal  result  is  stated 
in  the  following  terms. 

"  Proposition  XXV.  —  In  a  cycloid  whose  axis  is  vertical  and  whose 
summit  is  placed  below,  the  times  of  descent  in  which  a  particle  starts 
from  any  point  on  the  curve  and  reaches  the  lowest  point  are  equal  to 
each  other  ;  their  ratio  with  the  time  of  vertical  fall  along  the  whole  axis 
of  the  cycloid  is  equal  to  the  ratio  of  half  the  circumference  of  a  circle  to 
its  diameter.  " 

This  result  may  be  obtained  easily  by  means  of  a  well-known 
analysis.  Thus  the  motion  of  a  heavy  particle  on  a  cycloid  is  defined 
by  the  differential  equation 


dt2       4R 

Here  5  denotes  the  distance  of  the  particle  from  A  as  measured 
along  the  arc.     If  the  particle  starts  from  rest  at  the  point  J3,  and 

rr 

if  the  arc  AB  is  equal  to  $0,  it  follows  that  s  =  s0  cos  y  ~-  -  1.     The  sum- 

jt     /4iR 
mit,  A,  is  attained  in  a  time  T=  —  V/  —  •     Now  if  the  particle  were 

o 
1  Cf.  The  Complete  Works  of  Christiaan  Huyghens,  Vol.  XVIII,  pp.  152  to  184. 


186  THE   FORMATION   OF   CLASSICAL   MECHANICS 

allowed  to  fall  freely  along  DA,  it  would  reach  A  after_a  time  T  given 
by  2R  ==  i  gT'2,  from  which  it  foUows  that  T  =  V/ — •     Therefore  it 

<£  O 

must  be  that  —  =  -,  which  is  the  statement  that  Huyghens  makes.1 
T        2 


B 


Fig.  67 

We  shall  pass  over  the  third  part  of  Huyghens'  treatise,  which 
is  devoted  to  curved  lines.  The  question  is  that  of  the  search  for 
developments.  Huyghens  called  the  development  of  a  curve,  the 
evoluta,  and  the  development  the  descripta  ex  evolutions. 

Notable  among  this  work  is  the  development  of  the  cycloid,  which 
rationally  justified  the  use  of  the  cycloidal  shape  in  his  clock.  He 
also  studied  the  developments  of  conies — in  particular,  those  of  the 
parabola,  which  he  called  paraboloides.2 

7.   THE   THEORY   OF  THE    CENTRE   OF   OSCILLATION. 

"We  now  come  to  the  fourth  part  of  the  Horologium  oscillatorium, 
which  is  devoted  to  an  investigation  of  the  centre  of  oscillation. 

"  A  long  time  ago,  when  I  was  still  almost  a  child,  the  very  wise 
Mersenne  suggested  to  me,  and  to  many  others,  the  investigation  of 
centres  of  oscillation  or  agitation.  "  Thus  Huyghens  expresses  him 
self  at  the  beginning  of  the  fourth  part  of  his  major  work.  At  first 
he  found  nothing  "which  might  open  the  way  to  this  contemplation.  " 
However,  he  returned  to  the  question  in  order  to  improve  the  pendulums 
of  his  "  automatic,  "  to  which  he  had  been  led  to  add  moveable  weights 
above  the  principal  fixed  weight.  Huyghens  completely  resolved 

1  It  is  of  some  interest  to  remark  that  this  study  of  the  cycloid  was  very  popular 
among  XVIIth  Century  geometers.  WHEN  had  calculated  its  length,  ROBERVAL  had 
defined  the  tangents  while  PASCAL  determined  the  centre  of  gravity  and  calculated 
the  area.  WALLIS  too,  had  made  analogous  investigations. 

2  HUYGHENS  knew  the  development  of  a  parabola  as  early  as  1659,  as  a  result  of 
his  work  with  Jean  VAN  HEURAET  of  Harlem. 


WALLIS,   WREN,    HUYGHENS,   MARIOTTE 


187 


this  question  by  appealing  to  a  kind  of  generalisation  of  TorricellTs 
principle  that  depended  on  the  principle  of  living  forces. 

First  he  defines  the  compound  pendulum  and  the  centre  of  oscillation. 
The  latter  is  the  point  on  the  perpendicular  to  the  axis  of  oscillation 
through  the  centre  of  gravity  which  is  separated  from  the  axis  by  a 
distance  equal  to  the  length  of  the  simple  isochronous  pendulum. 

Huyghens    starts    from    the    following    fundamental    hypothesis. 

"  We  suppose  that  when  any  number  of  weights  starts  to  fall,  the 
common  centre  of  gravity  cannot  rise  to  a  height  greater  than  that  from 
which  it  starts.  " 

In  the  commentary  which  accompanies  this  hypothesis,  Huyghens 
specifies  that  verticals  should  be  considered  as  parallels  if  the  conside 
ration  of  a  centre  of  heaviness  is  to  have  any  meaning.  His  hypothesis 
reduces  to  the  following —  no  heavy  body  can  rise  by  the  sole  agency 
of  its  own  gravity  ;  what  is  true  for  a  single  body  is  also  true  for  bodies 
which  are  attached  to  each  other  by  rigid  rods. 


If,  now,  the  bodies  considered  are  no  longer  connected  to  each  other, 
they  nevertheless  have  a  common  centre  of  gravity,  and  it  is  this  which 
cannot  rise  spontaneously. 


188  THE   FORMATION   OF   CLASSICAL   MECHANICS 

"  Let  there  be  weights  A,  B,  C  and  let  D  be  their  common  centre 
of  gravity.  Suppose  the  horizontal  plane  is  drawn  and  that  EDF  is 
a  right  section  of  it.  Let  DA,  DB,  DC  be  the  rigid  lines  joining  the 
points  to  each  other  in  a  rigid  way.  Now  set  the  weights  in  motion 
so  that  A  comes  to  E  in  the  plane  EF.  Since  all  the  rods  are  turned 
through  the  same  angle,  B  will  now  be  at  G  and  C  at  H. 

"  Finally  suppose  that  B  and  C  are  joined  by  the  rod  HG  which 
cuts  the  plane  EF  in  F.  The  point  F  must  also  be  the  centre  of  gravity 
of  these  two  weights  taken  together,  since  D  is  the  centre  of  gravity  of 
the  three  weights  at  E,  G,  H  and  that  of  the  body  E  is  also  in  the  plane 
DEF.  The  weights  H  and  G  are  once  more  set  in  motion  about  the 
point  F  as  about  an  axis  and,  without  any  force,  simultaneously  arrive 
in  the  plane  EF.  Thus  it  appears  that  the  three  weights,  which  were 
originally  at  A,  B  and  C,  have  been  carried  exactly  to  the  height  of  their 
centre  of  gravity  by  their  own  equilibrium,  Q.  E.  D.  The  demonstra 
tion  is  the  same  for  any  other  number  of  weights. 

"  Now  the  hypothesis  that  we  have  made  is  also  applicable  to  liquid 
bodies.  By  its  means,  not  only  may  all  that  Archimedes  has  said  about 
floating  bodies  be  demonstrated,  but  also  many  other  theorems  in 
mechanics.  And  truly,  if  the  inventors  of  new  machines  who  strive  in 
vain  to  obtain  perpetual  motion  were  able  to  make  use  of  this  hypothesis, 
they  would  easily  discover  their  errors  for  themselves  and  would  under 
stand  that  this  motion  cannot  be  obtained  by  any  mechanical  means.  " 

Huyghens'  second  hypothesis  consists  of  the  neglect  of  the  resistance 
of  the  air  and  all  other  disturbances  of  motion. 

His  first  three  propositions  relate  to  the  geometry  of  masses.  We 
come  to  the  fourth. 

"  Proposition  IV.  —  If  a  pendulum  composed  of  several  weights,  and 
starting  from  rest,  has  executed  some  part  of  its  whole  oscillation,  and  it  is 
imagined  that,  from  that  moment  on,  the  common  bond  of  the  weights  is 
broken  and  that  each  of  the  weights  directs  its  acquired  velocity  upwards 
and  rises  to  the  greatest  height  possible,  then  by  this  means  the  common 
centre  of  gravity  will  rise  to  the  height  it  had  at  the  start  of  the  oscillation. 

"  Let  a  pendulum  composed  of  any  number  of  weights  A,  B,  C  be 
connected  by  a  weightless  rod  which  is  suspended  from  an  axis  D  per 
pendicular  to  the  plane  of  the  diagram.  The  centre  of  gravity,  E,  of 
the  weights  A,  B,  C  is  supposed  to  be  in  this  plane.  The  line  of  the 
centre,  DE,  makes  an  angle  EDF  with  DF  and  the  pendulum  is  drawn 
aside  as  far  as  this.  Suppose  that  it  is  released  in  this  position  and 
that  it  executes  a  part  of  its  oscillation  in  such  a  way  that  the  weights 
A,  B,  C  come  to  G,  H,  K.  Suppose  that  each  of  these  weights  directs 
its  velocity  upwards  when  the  bond  is  broken  (this  can  be  arranged  by 


WALLIS,   WREN,   HUYGHENS,   MARIOTTE 


189 


the  adjunction  of  certain  inclined  planes)  and  rises  to  the  greatest 
possible  height,  as  far  as  L,  M,  N.  Let  P  be  the  centre  of  gravity  of 
the  weights  when  they  have  attained  these  positions.  I  maintain  that 
this  point  is  at  the  same  height  as  E. 


R 


B 


Fig.  69 

"  First,  it  is  certain  that  P  is  not  higher  than  E  (hypothesis  I).     But 
neither  is  it  at  a  lesser  height.     Indeed,  if  this  is  possible,  let  P  be  lower 
than  E.     Suppose  that  the  weights  fall  down  p™*™^*?  ™ 
heights  that  they  travelled  in  mounting— namely  LG,  Mti,  I\^.     it 
clear  that  they  will  attain  the  same  velocity  as  they  had  at  the  beginning 
of  their  climb-that  is  to  say  the  velocity  they  acquired  m  the  motion 
of  the  pendulum  from  CBAD  to  KHGD.     Consequently   ^ jhey  are 
simultaneously  attached  again  to  the  rod  which  supported  ^em,  they 
will  continue  their  motion  along  the  arcs  winch  they  had  started  along 
(This  will  happen  if,  before  coming  to  the  rod    they  rebound  on  the 
planes  00.)     The  pendulum  reconstituted  in  this  way  will  effect  the 
rest  of  its  motion  without  any  interruption.     So  that  the  centre  of  gra 
vity  E  travels,  in  rising  and  falling,  along  the  equal  arcs  EF  and  *«, 
and  finds  itself  at  R-at  the  same  height  as  at  E.     But  we  have  supposed 
that  R  is  higher  than  P,  the  centre  of  gravity  of  the  weights  whe* .they 
are  at  L,  M,  N.    Therefore  R  will  also  be  higher  than  P.     The  centre 
of  gravity  of  the  weights  which  have  fallen  from  L  M,  N  will  herefore 
have  risen  by  a  height  greater  than  that  from  which  they  fell which 
is  absurd.     The  centre  of  gravity  P  is  not,  therefore,  lower  than  E. 


190 


THE   FORMATION   OF   CLASSICAL  MECHANICS 


No  more  is  it  at  a  greater  height.  It  must  therefore  be  that  it  is  at  the 
same  height.  Q.  E.  D.  " 

"  Proposition  V.  —  Being  given  a  pendulum  composed  of  any  number 
of  weights,  if  each  of  these  is  multiplied  by  the  square  of  the  distance  from 
the  axis  of  oscillation,  and  the  sum  of  these  products  is  divided  by  the 
product  of  the  sum  of  the  weights  with  the  distance  of  their  centre  of  gravity 
from  the  same  axis  of  oscillation,  there  will  be  obtained  the  length  of  the 
simple  pendulum  which  is  isochronous  with  the  compound  pendulum — that 
is,  the  distance  between  the  axis  and  the  centre  of  oscillation  of  the  com 
pound  pendulum.  " 

We  shall  analyse  this  demonstration  instead  of  reproducing  it.  Let 
A,  JB,  C  be  the  material  points  which  constitute  the  compound  and 
a,  ft,  c  be  their  weights.  Suppose  that  DA  =  e,  DB  =  /,  DC  =  g  and 
ED  =  d.  Also  suppose  that  E  is  the  centre  of  gravity  of  the  weights. 
Initially  the  compound  pendulum  is  released  from  rest  in  the  position 
DABC.  Let  FG  be  a  simple  pendulum  isochronous  with  the  compound 
pendulum  and  placed,  initially,  at  FG.  Let  the  angle  FGH  be  equal 
to  the  angle  EDF.  On  DE,  mark  oS  the  length  of  the  simple  pendulum, 


Fig.  70 

x  —  FG.  There  is  isochronism  between  the  simple  pendulum  and  the 
compound  pendulum  if,  at  corresponding  points  of  the  two  oscillations, 
0  and  P — such  that  the  arc  GO  is  equal  to  the  arc  LP — the  velocities 
of  G  and  L  are  equal. 


WALLIS,   WREN,   HUYGHENS,   MARIOTTE 
F 


191 


We  shall  show  that  this  equality  holds  for 


-     (a  +  b  +  C)d 

Indeed,  suppose  that  the  velocity  from  L  to  P  is  greater  than  that 
from  G  to  0. 

Let  SP,  RQ,  YO  be  the  descents  from  the  points  L,  JE,  G  to  the 
corresponding  points  P,  (),  0. 

d 
If  SP  =  y,  then  RQ  =  y  -. 

re 

The  simple  pendulum  G  has  a  velocity  at  0  which  is  sufficient  to 
enable  it  to  return  to  the  height  of  M,  either  along  OM  or  along  OY 
by  means  of  a  suitably  chosen  elastic  impact.  Then  the  point  L  has, 
at  P,  a  velocity  greater  than  that  which  would  enable  it  to  return  along 
PS  =  OY. 

Let  hL  be  greater  than  y,  the  height  to  which  L  can  return. 

The  points  A9  JS,  C  travel  the  arcs  AT,  BF,  CX  while  L  travels  LP. 
Thus  for  yi  there  obtains  the  relation 

v(A)  to  T       D.A  ^  e_ 

v(L)  to  P  ""  DL  ~~~  x 

Now  the  heights  of  return  are  proportional  to  the  squares  of  the 
velocities.  Therefore  the  height  of  return  of  A9  say  ft^,  is  greater  than 

e2 

—  y,  from  the  moment  that  L  exceeds  y. 


192  THE    FORMATION   OF   CLASSICAL   MECHANICS 

The  same  is  true  for  B  and  C,  so  that 


Now  this  inequality  expresses  the  fact  that  the  centre  of  gravity  E 
can  return  to  a  greater  height  than  the  one — RQ — from  which  it  fell, 

for  hE  = .     This  result  is  in  contradiction  with  pro- 

a  +  b  +  c  r 

position  IV  and  therefore  impossible.  In  the  same  way  the  hypothesis 
that  the  velocity  from  L  to  P  will  be  less  than  the  velocity  from  G  to  0 
implies  a  contradiction  with  the  same  proposition.  Therefore  the  pen 
dulum  FG  of  length  x  is  synchronous  with  the  compound  pendulum. 
This  establishes  the  required  result. 

We  shall  not  discuss  the  applications  of  these  propositions  here,  but 
shall  indicate  how  Huyghens  was  able  to  demonstrate  the  reciprocity 
between  the  axis  of  suspension  and  the  axis  of  oscillation.  Huyghens 
stated  the  following  proposition. 

"  Proposition  XVIII.  —  If  the  plane  space,  whose  product  with  the 
number  of  particles  of  the  suspended  magnitude  is  equal  to  the  sum  of  the 
squares  of  their  distances  from  the  axis  of  gravity.,  is  divided  by  the  distance 
between  the  two  axes,  the  result  obtained  is  the  distance  from  the  centre 
of  gravity  to  the  centre  of  oscillation.  " 

In  this  enunciation,  the  axis  of  gravity  is  the  axis  through  the  centre 
of  gravity  and  parallel  to  the  axis  of  suspension. 

yya 

Huyghens'  plane  space  has  the  value  •= — ,  where  r'  is  the  distance 

n 

from  the  axis  of  gravity  of  one  of  the  n  equal  particles  constituting  the 
suspended  magnitude.  It  is  therefore  identical  with  the  square  of  the 
radius  of  gyration  of  the  pendulum,  £2,  about  this  axis.  If  x  denotes 
the  length  of  the  simple  isochronous  pendulum  and  d  the  distance 
between  the  centre  of  gravity  and  the  axis  of  suspension,  Huyghens' 
statement  may  be  written 

nd          a 

V    2 

Now,  because  of  proposition  V,  the  length  x  is  equal  to          ,  where 

nd 
r  is  the  distance  from  one  of  the  n  particles  to  the  axis  of  suspension. 

This  is  equal  to  — ,  where  K  is  the  radius  of  gyration  of  the  pendulum 


WALLIS,    WREN,    HUYGHENS,    MARIOTTE  193 

ibout  this  axis.     Huyghens'  long  demonstration  of  proposition  XVIII 
•educes,  then,  to  the  verification  of  the  equality 


This  is  an  immediate  consequence  of  the  very  definition  of  the  mo 
ment  of  inertia,  which  was  to  he  introduced  by  Euler. 

Huyghens  then  states  the  following  proposition. 

66  Proposition  XIX.  —  When  the  same  magnitude  oscillates,  the  suspen 
sion  being  sometimes  shorter,  sometimes  longer,  the  distances  from  the 
centre  of  oscillation  to  the  centre  of  gravity  are  inversely  proportional  to 
the  distances  from  the  axes  of  suspension  to  the  centre  of  gravity.  " 

r™  •  .  .     i  -i  .        x  —  d         dr         .,   . 

Ihis  statement  is  equivalent  to  the  equation  —f  -  -  =  —  and  is  a 

xf  —  d'        d 

direct  consequence  of  proposition  XVIII. 

Finally,  Huyghens  was  able  to  state  the  reciprocity  of  the  two  axes, 

"  Proposition  XX.  —  The  centre  of  oscillation  and  the  point  of  sus 
pension  are  reciprocal.  " 

This  reciprocity  is  a  direct  consequence  of  Proposition  XIX  and  the 
constancy  of  the  product  d  (x  —  d). 

In  Huyghens'  work  the  whole  theory  of  the  centre  of  oscillation  rests 
on  the  fundamental  hypothesis  described  on  page  187.  This  is  equi 
valent  to  the  a  priori  assumption  of  the  conservation  of  living  forces. 
In  Lagrange's  opinion,  he  thus  sets  out  from  an  "  indirect  precept.  " 
Huyghens'  theory  produced  its  critics,  like  Roberval,  Catelan,  Jacques 
Bernoulli  and  others  —  but  these  had  nothing  more  than  "  evil  objec 
tions.  "  x  However,  criticism  had  the  value  of  attracting  the  attention 
of  geometers  —  as  did  the  efforts  of  Descartes  and  Roberval  to  solve  the 
same  problem  —  to  the  investigation  of  the  velocities  lost  or  gained  in 
the  constrained  motion  of  the  elementary  weights  that  constitute  a  com 
pound  pendulum.  Jacques  Bernoulli  was  at  first  mistaken  in  this 
investigation,  in  that  he  considered  the  velocities  acquired  in  a  finite 
time.  The  Marquis  de  1'Hospital  drew  his  attention  to  the  fact  that 
an  infinitesimal  motion  of  the  system  should  be  considered.  It  was  due 
to  this  remark,  made  in  1680,  that  Jacques  Bernoulli  arrived  at  a  new 
solution  of  the  problem  of  the  centre  of  oscillation  (1703).  We  shall 
return  to  a  discussion  of  this  solution,  which  prepared  the  way  for 
d'Alembert's  principle. 

1  Mfaanique  analytique,  Part  II,  Section  I. 

13 


194 


THE   FORMATION    OP   CLASSICAL   MECHANICS 


8.   THE  THEORY  OF  CENTRIFUGAL  FORCE. 

The  Horologium  oscillatorium  finishes  with  thirteen  unproved  propo 
sitions  on  centrifugal  force  and  the  conical  pendulum.  These  proposi 
tions  were  the  subject  of  De  vi  centrifuga.  This  manuscript  was  written 
in  1659  but  did  not  appear  until  1703,  in  a  form  that  was  edited  by  de 
Voider  and  Fullenius  and  published  posthumously  with  the  remainder 
of  Huyghens  work. 

In  this  treatise  Huyghens  regards  gravity  as  a  tendency  (conatus) 
to  fall.  This  tendency  is  made  apparent  by  the  tension  of  the  thread 
which  supports  a  body.  To  measure  it,  it  is  necessary  to  consider  the 
first  motion  of  the  body  after  the  thread  has  been  broken.  In  this  way 
the  conatus  is  caught  in  life,  before  there  has  been  time  for  it  to  have 
been  destroyed. 

Given  this,  Huyghens  sets  out  to  determine  the  conatus  of  a  body 
attached  to  a  revolving  wheel.  By  an  artifice  whose  object  is  clearly 
that  of  introducing  a  reference  system  bound  to  the  wheel,  he  assumes  that 
the  wheel  is  sufficiently  large  to  carry  a  man  who  is  attached  to  it.  This 
man  holds  a  thread,  supporting  a  ball  of  lead,  in  his  hand.  Because  of 
the  rotation  the  thread  is  stretched  with  the  same  force  as  if  it  were 
fixed  at  the  centre  of  the  wheel.  In  equal  times  the  man  travels  the 


0      6 


Fig.  72 


very  small  arcs  BE  and  B F.  If  it  is  released  at  B,  the  lead  will  travel 
along  the  rectilinear  paths  BC  and  CD  which  are  equal  to  these  arcs. 
The  points  C  and  D  do  not  fall  on  the  radii  AE  and  EF,  but  very  slightly 
behind  them. 


WALLIS,  WREN,   HUYGHENS,   MARIOTTE  195 

If  the  points  C  and  D  coincide  with  y  and  <?,  points  on  the  radii  AE, 
4F,  the  lead  will  tend  to  move  away  from  the  man  along  the  radius. 
The  distances  JEJy,  F<5,  .  . .  increase  as  the  square  numbers  1,  4,  9, 16, .  . . 
and  this  becomes  more  accurate  as  the  arcs  BE,  jEF, . . .  become  smaller. 
Now,  according  to  Galileo's  laws,  the  distances  travelled  by  a  body 
that  starts  its  fall  from  rest  increase  as  the  successive  square  numbers 
1,  4,  9,  16,  ...  The  conatus  which  is  sought  will  therefore  be,  on  this 
hypothesis,  the  same  as  that  of  a  heavy  body  suspended  by  a  thread. 

In  fact,  however,  the  points  C  and  D  lie  behind  y  and  d.  Therefore, 
with  respect  to  the  radius  on  which  it  is  placed,  the  weight  tends  to 
describe  a  path  which  is  tangential  to  the  radius.  But  at  the  moment 
of  the  separation  of  the  lead  and  the  wheel,  these  curves  can  be  regarded 
as  being  the  same  as  their  tangents  J5y,  F<3,  .  . .  with  the  consequence 
that  the  distances  EC,  FD,  .  .  .  must  be  considered  as  increasing  as 
the  series  1,  4,  9,  16,  .  .  .  And  here  is  Huyghens'  conclusion. 

"  The  conatus  of  a  sphere  attached  to  a  revolving  wheel  is  the 
same  as  if  the  sphere  tended  to  advance  along  the  radius  with  a  uni 
formly  accelerated  motion. ...  It  is  sufficient,  indeed,  that  this  motion 
should  be  observed  at  the  beginning.  Afterwards,  the  motion  can 
follow  every  other  law.  This  cannot  affect  the  conatus  that  exists  at 
the  beginning  of  the  motion  in  any  way.  This  conatus  is  entirely 
similar  to  that  of  a  body  hung  by  a  thread.  From  which  we  conclude 
that  the  centrifugal  forces  of  unequal  particles  that  move  with  equal 
velocities  on  equal  circles  have  the  same  relation  to  each  other  as  their 
gravities,  that  is,  as  their  quantities  of  solid  (quantitates  solidae) .  [Here 
we  catch  a  fleeting  glimpse  of  the  concept  of  mass.]  Indeed,  all  bodies 
tend  to  fall  with  the  same  velocity  in  the  same  uniformly  accelerated 
motion.  But  their  conatus  has  a  moment  (momentum)  that  is  greater 
as  the  bodies  themselves  are  greater.  It  must  be  the  same  for  bodies 
that  tend  to  move  away  from  a  centre,  since  their  conatus  is  similar 
to  that  which  arises  from  their  gravity.  But  while  the  same  sphere 
always  has  the  same  tendency  to  fall  whenever  it  is  hung  from  a  thread, 
the  conatus  of  a  sphere  attached  to  a  revolving  wheel  depends  on  the 
velocity  of  rotation  of  the  wheel.  It  remains  to  us  to  find  the  magnitude 
or  the  quantity  of  conatus  for  different  velocities  of  the  wheel.  "  l 

So  much  for  the  principle  of  centrifugal  force  that  Huyghens 
developed  in  his  preamble.  We  shall  now  state  the  propositions 
that  he  established  in  a  much  abbreviated  form. 

1)  For  a  given  period  of  rotation,  the  centrifugal  force  is  propor 
tional  to  the  diameter. 

1  The  Complete  Works  of  Christiaan  Huyghens,  Vol.  XVI,  p.  266. 


196 


THE    FORMATION   OF    CLASSICAL   MECHANICS 


2)  For  a  given  velocity  on  the  circumference,  it  is  inversely  propor 
tional  to  the  diameter. 

3)  For  a  given  radius,  it  is  proportional  to  the  square  of  the  velocity 
on  the  circumference. 

4)  For  a  given  centrifugal  force,  the  period  of  revolution  is  propor 
tional  to  the  square  root  of  the  radius. 

5)  "  When  a  particle  moves  on  the  circumference  with  the  velocity 
that  it  would  have  acquired  in  falling  from  a  height  equal  to  a  quarter 
of  the  diameter,  its  centrifugal  force  is  equal  to  its  gravity.     In  other 
words,  it  will  stretch  the  cord  to  which  it  is  attached  with  the  same  force 
as  if  it  were  suspended.  " 

We  shall  summarise  Huyghens'  proof  of  this  proposition. 


The  particle  is  supposed  to  describe  the  circumference  of  a  circle 
with  uniform  motion  and  the  velocity  (\/Kg)   which  it  would  have 

T) 

acquired  in  falling  from  the  height  CJB  =  — .     If  it  is  detached  at  B,  it 

2i 

travels  along  the  tangent  uniformly,  covering  a  distance  BD  =  R  in  the 
time  I  i/  __  J  which  it  would  have  spent  in  falling  along  CB.    We  consider 

\        o  / 

a  very  small  fraction  of  BD — namely  BE — and  draw  the  straight  line 

C  C*      /  R  T?\  ^ 
EFAH .     We  also  suppose  that  _==(_).    Then  BE,  or  ( <\/2R.CG) , 

Ojt>       \  JoXx  / 

is  proportional  to  the  time  of  free  fall  along  CG,  which  is  equal  to 


V1- 


8 


WALLIS,   WREN,   HUYGHENS,   MARIOTTE 


197 


The  particle  detached  at  B  travels  the  distance  EE  uotformly 
in  the  time  it  would  have  spent  in  falling  freely  from  the  height  CG. 
Now  BE  can  be  approximated  to  by  the  arc  BF.  If  it  is  shown  that 
CG^FE  it  will  have  been  proved  that  the  conatus  of  the  centrifugal 
force  is  equal  to  the  conatus  of  the  gravity,  for  the  particle  considered. 


-  B£2  =  —  i—\\=  CG.  The  proposition  is  etablished. 

—   2R        2  \  JR  /  *• 

It  must  be  pointed  out  that  for  Huyghens  centrifugal  force  is  in  no 
way  a  fictitious  force.  On  the  contrary,  he  attributed  to  it  both  measure 
ment  and  special  privilege  by  identifying  it  with  gravity  in  the  part- 
icular  case  we  have  just  seen. 

Let  us  continue  our  examina 
tion  of  the  propositions  that  end 
the  Horologium  oscillatorium  : 

6)  A  conical  pendulum  is  iso 
chronous  with  a  simple  pendulum 
having  as    length   half  the   latus 
rectum    (parameter)     of   the    par 
aboloid   of  which    it    describes    a 
parallel. 

7)  The  period  of  a  conical  pen 
dulum  depends  only  on  the  height 
of  the  cone. 

8)  It    is    proportional   to    the 
square  root  of  this  height. 

9)  The  period  of  motion  of  a 
conical  pendulum  "  on  extremely 
small  circumferences  "  is  equal  to 
n  T,  T  being  the  time  it  falls  freely 
from  a  height  twice  the  length  of 
the  pendulum. 

10)  If  a  mobile  runs  along  a 
circumference  and  if  the  period  of 
its  uniform  motion  is  equal  to  the 
time  in  which  a  conical  pendulum, 

whose  length  is  equivalent  to  the  •  «.  •*       i 

radius,   describes   an   extremely   small   circumference,  its   centrifugal 
force  is  equal  to  its  gravity.  . 

11)  The  period  of  revolution  of  a  conical  pendulum  is  equal  to  the 
time  it  takes  to  fall  freely  from  a  height  equal  to  its  length,  if  the  string 


Fig.  74 


198  THE    FORMATION   OF   CLASSICAL   MECHANICS 

forms  with  the  horizontal  plane  an  angle  the  sine  of  which  is  equal  to 


12)  The  tension  of  the  string  of  a  conical  pendulum  of  given  height 
is  proportional  to  the  length  of  the  pendulum. 

13)  When  a  simple  pendulum  performs  a  maximum  lateral  motion, 
that  is,  when  it  descends  by  a  whole  quarter  of  the  circumference,  the 
tension  of  the  string  at  the  lowest  point  is  three  times  the  weight  of  the 
pendulum.1 

Without  spending  time  over  the  demonstrations  of  these  propositions, 
we  may  mention,  for  the  sake  of  curiosity,  a  clock  constructed  by 
Huyghens,  which  illustrates  this  theory. 

The  axis  KH  is  vertical,  the  curved  line  A  I  is  the  e  volute  of  the 
parabola  FEC.  In  the  rotation  of  the  axis,  the  pendulum  BF,  which 
escapes  tangentially  from  the  evolute,  describes  a  parallel,  a  straight 
section  of  the  paraboloid  engendered  by  FEC. 


9.   HUYGHENS   AND   THE   PRINCIPLE    OF   RELATIVITY. 

In  Huyghens'  sense,  the  principle  of  relativity  is  that  it  is  impossible 
that  an  observer  in  uniform  rectilinear  motion  can  discover  his  own 
translation.  We  have  seen  how  Huyghens  exploited  this  principle 
in  his  study  of  the  laws  of  impact. 

Huyghens  appears  to  have  assumed,  however,  in  his  work  on  the 
centrifugal  force,  the  tangible  reality  of  uniform,  circular  motion, 
which  he  called  motus  verus.  Nevertheless,  he  went  back  on  this  opinion 
after  the  appearence  of  Newton's  Principia.  In  the  last  analysis, 
he  rejected  the  concept  of  absolute  motion  and  remained  a  Cartesian. 
Indeed,  in  a  fragment  of  his  writing  that  must  be  placed  later  than 
1688,  he  wrote,  "  In  circular  motion  as  well  as  in  straight  and  free 
motion  there  is  nothing  that  is  not  relative,  in  the  sense  that  this 
is  all  there  is  to  know  of  motion.  " 

We  do  not  have  the  time  to  deal  with  Huyghens'  contribution  to 
physics.  We  only  recall  that  he  outlined  an  undulatory  theory  of  light 
in  1673 — he  had  learnt  of  Erasme  Bartholin's  experimental  discovery 
of  the  birefringence  of  Iceland  Spar  in  1670,  and  sought  to  find  a  rational 
explanation  of  this  phenomenon.  Huyghens  read  his  Treatise  on  Light 
before  the  Academic  des  Sciences  at  Paris  in  1690. 

1  (Euvres  completes  de  Huyghens,  vol.  XVI,  p.  280  sq. 


WALLIS,    WREN,   HUYGHENS,   MARIOTTE 


199 


10.  MARIOTTE  AND  THE  LAWS  OF  IMPACT. 

Mariotte  dealt  with  the  theory  of  impact  in  his  Treatise  on  the 
percussion  or  impact  of  bodies. 

This  work  added  nothing  essentially  novel  to  the  work  of  Wallis, 
Wren  and  Huyghens,  but  was  distinguished  by  its  much  more  expe 
rimental  approach  to  the  subject.  Mariotte  rejected  the  concept  of 
perfectly  hard  bodies  in  the  sense  that  "Wallis  had  used  the  term.  He 
confined  himself  to  bodies  that  were  flexible  and  resilient  (that  is, 
perfectly  elastic)  and  bodies  that  were  flexible  and  not  resilient  (that  is, 
perfectly  soft). 

In  order  to  obtain  velocities  of  direct  impact  that  were  in  any 
given  relationship,  Mariotte  described  an  apparatus  consisting  of  two 
equal  pendulums  which  were  allowed  to  fall  from  positions  J3J', 
EL'  that  could  be  chosen  at  will. 

Mariotte  had  the  merit  of  recognis 
ing  the  part  played  by  the  mass  in 
the  laws  of  impact.  Thus  he  wrote — 

"  The  weight  of  a  body  is  not 
understood,  here,  to  be  the  tendency 
which  makes  it  move  towards  the  centre 
of  the  Earth,  but  rather  to  be  its 
volume  together  with  a  certain  solidity 
or  condensation  of  the  parts  of  its 
material  which  is  probably  the  cause 
of  its  heaviness.  " 

Mariotte   was    also    concerned    with 

the  investigation  of  centres  of  percussion.  He  combined  the  laws  of 
statics  with  those  of  impact  in  this  investigation.  He  investigated 
the  percussions  exerted  on  a  balance  by  jets  of  water  of  known  amount 
and  came  to  the  following  conclusion.  "  Two  bodies  that  fall  on  a 
balance,  on  one  side  and  the  other,  are  in  equilibrium  at  the  moment 
of  impact  if  the  distances  [from  the  centre]  of  the  points  where  they 
fall  are  in  reciprocal  proportion  to  their  quantities  of  motion.  " 


Fig.  75 


CHAPTER   SIX 

NEWTON 

(1642-1727) 


1.   THE    NEWTONIAN    METHOD. 

Thanks  to  Galileo  and  Huyghens,  mechanics  had  been  emancipat 
ed  from  the  scholastic  discipline.  Essential  problems  like  the  motion 
of  projectiles  in  the  vacuum  and  the  oscillations  of  a  compound  pendulum 
had  been  solved.  Nevertheless,  the  task  of  constructing  an  organised 
corpus  of  principles  in  dynamics  remained.  This  was  the  work  of 
Newton,  who  set  his  seal  on  the  foundation  of  classical  mechanics 
at  the  same  time  that  he  extended  its  field  of  application  to  celestial 
phenomena. 

Newton's  work  in  mechanics  is  called  Philosophiae  naturalis  prin- 
cipia  mathematica  (1687) -1  It  proceeds  by  a  method  that  is  at  once 
rational  and  experimental,  to  which  the  author  himself  gave  us  the 
key. 

A  first  rule  of  the  newtonian  method  consists  in  not  assuming 
other  causes  than  those  which  are  necessary  to  explain  the  phenomena. 
A  second  is  to  relate  as  completely  as  possible  analogous  effects  to 
the  same  cause.  A  third,  to  extend  to  all  bodies  the  properties  which 
are  associated  with  those  on  which  it  is  possible  to  make  experiments. 
A  fourth,  to  consider  every  proposition  obtained  by  induction  from 
observed  phenomena  to  be  valid  until  a  new  phenomenon  occurs  and 
contradicts  the  proposition  or  limits  its  validity. 

It  was  by  relying  on  the  third  of  these  rules  that  Newton  was 
able  to  formulate  the  law  of  universal  gravitation.  In  expressing 
this  law,  Newton  had  no  intention  of  assigning  a  cause  to  gravitation. 
"  But  hitherto  I  have  not  been  able  to  discover  the  cause  of  those  pro 
perties  of  gravity  from  phenomena,  and  I  frame  no  hypotheses  (hypo- 

1  The  manuscript  of  the  Principia  was  deposited  with  the  Royal  Society  on  April 
28th,  1686.  It  was  published  for  the  first  time  in  1687,  on  the  intervention  of  HALLEY. 


NEWTON  201 

theses  non  fingo).  For  whatever  is  not  deduced  from  the  phenomena, 
is  to  be  called  an  hypothesis  ;  and  hypotheses,  whether  metaphysical 
or  physical,  whether  of  occult  qualities  or  mechanical,  have  no  place 
in  experimental  philosophy.  In  this  philosophy  particular  propo 
sitions  are  inferred  from  the  phenomena,  and  afterwards  rendered 
general  by  induction.  "  1 

It  is  not  surprising  that  Newton  himself  should  have  departed 
from  this  rule  and  that  he  should  have  introduced  purely  abstract 
entities  into  some  of  his  arguments.  But  on  the  whole,  his  work  is 
a  practical  expression  of  the  natural  philosophy  whose  foundation 
he  laid. 

2.   THE    NEWTONIAN   CONCEPTS. 

Newton  introduced  the  notion  of  mass  into  mechanics.  This 
notion  had  appeared  in  Huyghen's  work,  but  only  in  an  impermanent 
form.2 

"  Definition  /.  —  The  Quantity  of  Matter  is  the  measure  of  the  same, 
arising  from  its  density  and  bulk  conjunctly. 

"...  I  have  no  regard  in  this  place  to  a  medium,  if  such  there 
is,  that  freely  pervades  the  interstices  between  the  parts  of  bodies. 
It  is  this  quantity  that  I  mean  hereafter  every  where  under  the  name 
of  Body  or  Mass.  And  the  same  is  known  by  the  weight  of  each  body — 
For  it  is  proportional  to  the  weight,  as  I  have  found  by  experiments 
on  pendulums,  very  accurately  made.  ..." 

In  these  experiments  Newton  worked  with  pendulums  made  of 
different  materials  and  of  the  same  length,  and  established  that  their 
acceleration  did  not  depend  on  the  nature  of  the  material.  He  eliminat 
ed  the  variations  of  the  resistance  of  the  air  by  using  pendulums 
formed  of  spheres  of  the  same  diameter,  suitably  hollowed-out  to 
ensure  equality  of  the  weights. 

When  Newton  declared  that  the  mass  was  known  by  the  weight, 
he  contemplated  the  weight  in  a  given  place.  For  he  was  well  aware 
of  the  fact  that  the  weight  of  a  body  varied  with  its  distance  from 
the  centre  of  the  Earth,  while  its  mass  remained  constant. 

This  Newtonian  definition  of  the  mass  has  been  often  and  justly 
criticised.  Thus  Mach  wrote,  "  The  vicious  circle  is  clear,  since  the 
density  can  only  be  defined  as  the  mass  of  unit  volume.  Newton  clearly 
believed  that  to  each  body  was  associated  a  characteristic  determinant 

1  In  the  English  translation  of  the  present  book  quotations  from  the  work  of  NEWTON 
are  taken  from  Andrew  MOTTE'S  translation  of  the  Principia  (1724). 

2  See  above,  p.  195. 


202  THE   FORMATION   OF   CLASSICAL   MECHANICS 

of  its  motion,  which  was  different  from  its  weight  and  which  we,  with 
him,  call  mass.  But  he  did  not  succeed  in  expressing  this  idea  correctly." l 

"  Definition  II.  —  The  Quantity  of  Motion  is  the  measure  of  the 
same,  arising  from  the  velocity  and  quantity  of  matter  conjunctly. 

"  Definition  III.  —  The  vis  insita,  or  Innate  Force  of  Matter,  is 
a  power  of  resisting,  by  which  every  body,  as  much  as  in  it  lies,  endeavours 
to  persevere  in  its  present  state,  whether  it  be  of  rest,  or  of  moving  uniformly 
forward  in  a  right  line.  " 

To  Newton,  this  vis  insita  was  always  proportional  to  the  quantity 
of  matter.  He  also  gave  it — with  a  meaning  different  from  that  which 
is  accepted  now — the  name  of  force  of  inertia.  This  force  is  resistive 
when  it  is  desired  to  change  a  body's  state  of  motion,  and  impulsive 
to  the  extent  that  a  body  in  motion  acts  on  an  obstacle. 

"  Definition  IV.  —  An  impressed  force  (vis  impressa)  is  an  action 
exerted  upon  a  body,  in  order  to  change  its  state,  either  of  rest,  or  of  moving 
uniformly  forward  in  a  right  line.  " 

Therefore,  to  Newton,  the  vis  impressa  is  the  determinant  of  the  accel 
eration.  "  This  force  consists  in  the  action  only ;  and  remains 
no  longer  in  the  body,  when  the  action  is  over.  For  a  body  maintains 
every  new  state  it  acquires,  by  its  vis  insita  only.  "  The  impressed 
force  acts  by  impact,  pressure  or  at  a  distance. 

"  Definition  V.  —  A  Centripetal  force  is  that  by  which  bodies  are 
drawn  or  impelled,  or  any  way  tend,  towards  a  point  as  to  a  centre.  " 

As  examples  of  centripetal  force,  Newton  cites  the  gravity  which 
makes  bodies  tend  towards  the  centre  of  the  Earth,  the  magnetic 
force  that  attracts  iron  towards  a  magnet,  and  that  force — whatever 
its  nature  might  be — that  makes  each  planet  describe  a  curved  orbit. 

The  force  exerted  by  a  hand  that  whirls  a  stone  in  a  sling  is  also  a 
centripetal  force.  Newton  adds,  "  And  the  same  thing  is  to  be  under 
stood  of  all  bodies,  revolved  in  any  orbits  ;  and  were  it  not  for  the 
opposition  of  a  contrary  force  which  restrains  them  to,  and  detains 
them  in  their  orbits,  which  I  therefore  call  Centripetal,  would  fly  off 
in  right  lines,  with  an  uniform  motion.  " 

Newton  then  distinguishes  the  absolute  quantity,  the  accelerative 
quantity  and  the  motive  quantity  of  the  centripetal  force  (Definitions 
VI,  VII  and  VIII). 

The  absolute  quantity  depends  on  the  efficacy  of  the  cause  that 
propagates  the  centripetal  force — for  example,  the  size  of  a  stone  or  the 
strength  of  a  magnet. 

The  accelerative  quantity  is  measured  by  the  velocity  produced  in 

1  M.,  p.  190. 


NEWTON  203 

a  given  time.     Therefore,  in  modern  language,  it  is  the  acceleration 
produced  by  the  force. 

Newton  takes  the  value  of  the  motive  quantity  to  be  the  quantity 
of  motion  produced  in  a  given  time.  Therefore  it  is  the  motive  quantity 
which  satisfies  the  law  that  is  now  written — 

(1)  F  =  my. 

For  heavy  bodies,  the  motive  quantity  becomes  identified  with  the 
weight. 

In  this  way  Newton  multiplied  the  definitions  and  concepts.  Instead 
of  deducing  the  concept  of  motive  force  from  the  concepts  of  mass  and 
acceleration  by  using  the  law  (1),  he  consciously  regarded  the  mass  and 
the  force  as  two  primarily  distinct  notions. 

Newton  also  took  certain  precautions  in  order  to  anticipate  the 
objections  of  the  cartesian  philosophy,  and  to  make  the  notion  of  action 
at  a  distance  acceptable.  "  I  likewise  call  Attractions  and  Impulses, 
in  the  same  sense,  Accelerative  and  Motive ;  and  use  the  words  Attrac 
tion,  Impulse  or  Propensity  of  any  sort  towards  a  centre,  promiscuously, 
and  indifferently,  one  for  another ;  considering  those  forces  not  Physi 
cally  but  Mathematically —  Wherefore  the  reader  is  not  to  imagine,  that 
by  those  words  I  any  where  take  upon  me  to  define  the  kind,  or  the 
manner  of  any  Action,  the  causes  and  the  physical  reason  thereof,  or 
that  I  attribute  Forces,  in  a  true  and  Physical  sense,  to  certain  centres 
which  are  only  mathematical  points.  " 

Newton  then  proceeds  to  discuss  the  currently  used  concepts  of 
time,  space,  place  and  motion.  He  introduces  a  distinction  between 
the  relative,  apparent  and  common  senses  of  the  words  and  the  absolute, 
true  and  mathematical  senses. 

"  I,  Absolute,  true  and  mathematical  time,  of  itself,  and  from  its 
own  nature  flows  equably  without  regard  to  any  thing  external  and  by 
another  name  is  called  duration —  relative,  apparent  and  common  time 
is  some  sensible  and  external  (whether  accurate  or  unequable)  measure 
of  duration  by  the  means  of  motion,  which  is  commonly  used  instead 
of  true  time  ;  such  as  an  hour,  a  day,  a  month,  a  year. 

66 II.  Absolute  space,  in  its  own  nature,  without  regard  to  anything 
external,  remains  always  familar  and  immoveable.  Relative  space  is 
some  moveable  dimension  or  measure  of  the  absolute  spaces  ;  which  our 
senses  determine,  by  its  position  to  bodies  ;  and  which  is  vulgarly  taken 
for  immoveable  space. ..." 

Or  again,  "  It  may  be,  that  there  is  no  such  thing  as  an  equable 
motion,  whereby  time  may  be  accurately  measured.  All  motions  may 


204  THE   FORMATION   OF    CLASSICAL   MECHANICS 

be  accelerated  and  retarded,  but  the  true,  or  equable  progress,  of  abso 
lute  time  is  liable  to  no  change.  .  .  . 

"  For  times  and  spaces  are,  as  it  were,  the  places  as  well  of  themselves 
as  of  all  other  things.  All  things  are  placed  in  time  as  to  order  of 
succession  ;  and  in  space,  as  to  order  of  situation.  It  is  from  their 
essence  or  nature  that  they  are  places  ;  and  translations  out  of  those 
places,  are  the  only  absolute  motions.  " 

Newton  concerns  himself  with  distinguishing  absolute  and  relative 
motions  by  their  causes  and  their  effects. 

"  The  causes  by  which  true  and  relative  motions  are  distinguished, 
one  from  the  other,  are  the  forces  impressed  upon  bodies  to  generate 
motion.  True  motion  is  neither  generated  nor  altered,  but  by  some 
force  impressed  on  the  body  moved —  but  the  relative  motion  may  be 
generated  or  altered  without  any  force  impressed  upon  the  body.  .  .  . 

"  The  effects  which  distinguish  absolute  from  relative  motion,  are 
the  force  of  receding  from  the  axe  of  circular  motion.  For  there  are 
no  such  forces  in  a  circular  motion,  purely  relative,  but  in  a  true  and 
absolute  circular  motion,  they  are  greater  or  less,  according  to  the  quan 
tity  of  motion. 

"  If  a  vessel,  hung  by  a  long  cord,  is  so  turned  about  that  the  cord 
is  strongly  twisted,  then  filled  with  water,  and  held  at  rest  together  with 
the  water  ;  after  by  the  sudden  action  of  another  force,  it  is  whirled 
about  the  contrary  way,  and  while  the  cord  is  untwisting  itself,  the 
vessel  continues  for  some  time  in  this  motion  ;  the  surface  of  the  water 
will  at  first  be  plain,  as  before  the  vessel  began  to  move —  but  the  vessel, 
by  gradually  communicating  its  motion  to  the  water,  will  make  it  begin 
sensibly  to  revolve,  and  recede  by  little  and  little  from  the  middle,  and 
ascend  to  the  sides  of  the  vessel,  forming  itself  into  a  concave  figure 
(as  I  have  experienced)  and  the  swifter  the  motion  becomes,  the  higher 
will  the  water  rise,  till  at  last,  performing  its  revolutions  in  the  same 
times  with  the  vessel,  it  becomes  relatively  at  rest  in  it.  This  ascent 
of  the  water  shows  its  endeavour  to  recede  from  the  axe  of  its  motion  ; 
and  the  true  and  absolute  circular  motion  of  the  water,  which  is  here 
directly  contrary  to  the  relative,  discovers  itself,  and  may  be  measured  by 
this  endeavour.  At  first,  when  the  relative  motion  of  the  water  in  the 
vessel  was  greatest  it  produced  no  endeavour  to  recede  from  the  axe — 
the  water  shewed  no  tendency  to  the  circumference,  nor  any  ascent 
towards  the  sides  of  the  vessel,  but  remained  of  a  plain  surface,  and 
therefore  its  true  circular  motion  had  not  yet  begun.  But  afterwards, 
when  the  relative  motion  of  the  water  had  decreased,  the  ascent  thereof 
towards  the  sides  of  the  vessel,  proved  its  endeavour  to  recede  from  the 
axe  ;  and  this  endeavour  shewed  the  real  circular  motion  of  the  water 


NEWTON  205 

perpetually  increasing,  till  it  had  acquired  its  greatest  quantity,  when 
the  water  rested  relatively  in  the  vessel.  ..." 

Again  Newton  stresses  the  distinction  between  absolute  and  relative 
quantities.  "  And  if  the  meaning  of  the  words  is  to  be  determined  by 
their  use  ;  then  by  the  names  time,  space,  place  and  motion,  their 
measures  are  properly  to  be  understood  ;  and  the  expression  will  be 
unusual,  and  purely  mathematical,  if  the  measured  quantities  themselves 
are  meant.  Upon  this  account,  they  do  strain  the  Sacred  Writings,  who 
there  interpret  those  words  for  the  measured  quantities.  Nor  do  those 
less  defile  the  purity  of  Mathematical  and  Philosophical  truths,  who 
confound  real  quantities  themselves  with  their  relations  and  vulgar 
measures,  " 

Newton  did  not  conceal  the  difficulty  of  distinguishing  true  from 
apparent  motions,  because  "  the  parts  of  that  immoveable  space  in 
which  those  motions  are  performed,  do  by  no  means  come  under  the 
observation  of  our  senses.  "  In  order  to  accomplish  this  it  is  necessary, 
according  to  him,  to  make  use  simultaneously  of  the  apparent  motions, 
u  which  are  the  differences  of  the  true  motions  "  and  the  forces,  "  which 
are  the  causes  and  the  effects  of  the  true  motions.  " 

As  an  example  he  cites  the  motion  of  two  spheres  attached  by  an 
inflexible  thread  and  turning  about  their  centre  of  gravity.  The  tension 
of  the  thread  allows  "  the  quantity  of  circular  motion  "  to  be  measured. 

To  Newton,  force  therefore  appears  as  a  true  or  absolute  element  and 
is  opposed  to  motion,  which  only  has  a  relative  character  with  respect 
to  a  suitably  chosen  reference  system.  Certain  modern  critics — notably 
Mach — reproach  the  Newtonian  philosophy  for  its  metaphysical  cha 
racter  in  this  connection.  Absolute  space  and  time  appear  to  them  as 
purely  abstract  entities  which  can  only  be  deduced  from  observation. 
More  correctly,  theoretical  physics  is  based  on  the  introduction  of  pure 
unobservables  as  intermediaries  in  the  calculation.  Under  a  cloak  of 
metaphysical  appearance  it  contains  a  profound  physical  truth.  It 
"  explicitly  proclaims  to  the  student  of  mechanics  the  necessity  of 
considering  the  privileged  reference  frames  in  time  and  space  and  of  thus 
avoiding  the  confusion  that  is  so  apparent  in  the  ideas  of  Descartes  and 
Huyghens."1 

3.    THE   NEWTONIAN  LAWS  OF  MOTION. 

Newton  stated  the  principle  of  inertia  at  the  beginning.     This  had 
already  been  discovered  by  Galileo  and  reformulated  by  Huyghens. 
"  Law  L  —  Every  body  perseveres  in  its  state  of  rest,  or  of  uniform 

1  JOUGUET,  L.  M.,  Vol.  II,  p.  11,  note  9. 


206  THE   FORMATION  OF   CLASSICAL  MECHANICS 

motion  in  a  right  line,  unless  it  is  compelled  to  change  that  state  by  forces 
impressed  thereon. " 

He  next  repeats  the  idea  that  the  motive  force  is  the  determinant  of 
acceleration. 

"  Law  II.  —  The  alteration  of  [the  quantity  of]  motion  is  ever  propor 
tional  to  the  motive  force  impressed  ;  and  is  made  in  the  direction  of  the 
right  line  in  which  that  force  is  impressed, " 

The  third  law  constitutes  the  principle  of  the  equality  of  the  action 
and  the  reaction. 

"  Law  III.  —  To  every  action  there  is  always  opposed  an  equal 
reaction —  or  the  mutual  actions  of  the  two  bodies  upon  each  other  are 
always  equal,  and  directed  to  contrary  parts. 

"  Whatever  draws  or  presses  another  is  as  much  drawn  or  pressed 
by  that  other.  If  you  press  a  stone  with  your  finger,  the  finger  is  also 
pressed  by  the  stone.  If  a  horse  draws  a  stone  tyed  to  a  rope,  the 
horse  (if  I  may  so  say)  will  be  equally  drawn  back  towards  the  stone — 
For  the  distended  rope,  by  the  same  endeavour  to  relax  or  unbend  it 
self,  will  draw  the  horse  as  much  towards  the  stone,  as  it  does  the  stone 
towards  the  horse,  and  will  obstruct  the  progress  of  the  one  as  much 
as  it  advances  that  of  the  other. 

"  If  a  body  impinge  on  another,  and  by  its  force  change  the  motion 
of  the  other,  that  body  also  (because  of  the  equality  of  the  mutual 
pressure)  will  undergo  an  equal  change,  in  its  own  motion,  towards  the 
contrary  part. 

"  The  changes  made  by  these  actions  are  equal,  not  in  the  velocities, 
but  in  the  [quantities  of]  motion  of  the  bodies  ;  that  is  to  say,  if  the 
bodies  are  not  hindered  by  any  other  impediments.  For  because  the 
motions  are  equally  changed,  the  changes  of  velocity  made  towards 
contrary  parts,  are  reciprocally  proportional  to  the  bodies.  This  law 
also  takes  place  in  attractions.  ..." 

It  is  interesting  to  see  how  Newton — contrary  to  the  custom  of  the 
time — pays  homage  to  his  predecessors. 

"  Hitherto  I  have  laid  down  such  principles  as  have  been  received 
by  all  mathematicians,  and  are  confirmed  by  abundance  of  experiments. 
By  the  two  first  Laws  and  the  first  two  Corollaries,  Galileo  discovered 
that  the  descent  of  bodies  observed  the  duplicate  ratio  of  the  time,  and 
that  the  motion  of  projectiles  was  in  the  curve  of  a  parabola  ;  expe 
rience  agreeing  with  both,  unless  so  far  as  these  motions  are  a  little 
retarded  by  the  resistance  of  the  air.  .  .  . 

"  On  these  same  laws  and  corollaries  depend  those  things  which 
have  been  demonstrated  concerning  the  times  of  vibration  of  pendulums, 
and  are  confirmed  by  the  daily  experiments  of  pendulum  clocks.  By 


NEWTON  207 

the  same  together  with  the  third  Law  Sir  Christopher  Wren,  Dr.  Wallis 
and  Mr.  Huyghens,  the  greatest  geometers  of  our  times,  did  severally 
determine  the  rules  of  the  congress  and  reflection  of  hard  bodies,  and 
much  about  the  same  time  communicated  their  discoveries  to  the  Royal 
Society,  exactly  agreeing  among  themselves,  as  to  those  rules, 
Dr.  Wallis  indeed  was  something  more  early  in  the  publication  ;  then 
followed  Sir  Christopher  Wren,  and  lastly,  Mr.  Huyghens.  But  Sir 
Christopher  Wren  confirmed  the  truth  of  the  things  before  the  Royal 
Society  by  the  experiment  of  pendulums,  which  Mr.  Mariotte  soon  after 
thought  fit  to  explain  in  a  treatise  entirely  upon  that  subject.  " 

For  his  part,  Newton  repeated  the  experiments  with  great  care. 
From  them  he  concluded  that  "  the  quantity  of  motion,  collected  from 
the  sum  of  the  motions  directed  towards  the  same  way,  or  from  the 
difference  of  those  that  were  directed  towards  contrary  ways,  was  never 
changed,  "  whether  the  bodies  were  hard  or  soft,  elastic  or  not. 

In  order  to  justify  the  equality  of  action  and  reaction  in  the  case  of 
attractions,  Newton  argued  in  the  following  way. 

"  Suppose  an  obstacle  is  interposed  to  hinder  the  congress  of  any 
two  bodies  A,  B,  mutually  attracting  one  the  other —  then  if  either  body 
as  A,  is  more  attracted  towards  the  other  body  jB,  than  that  other 
body  B  is  towards  the  first  body  A,  the  obstacle  will  be  more  strongly 
urged  by  the  pressure  of  the  body  A  than  by  the  pressure  of  the  body  B  ; 
and  therefore  will  not  remain  in  aequilibrio —  but  the  stronger  pressure 
will  prevail,  and  will  make  the  system  of  the  two  bodies,  together  with 
the  obstacle,  to  move  directly  towards  the  parts  on  which  B  lies  ;  and 
in  free  spaces,  to  go  forward  in  infinitum  with  a  motion  perpetually 
accelerated.  Which  is  absurd,  and  contrary  to  the  first  law.  For  by 
the  first  law,  the  system  ought  to  persevere  in  it's  state  of  rest,  or  of 
moving  uniformly  forward  in  a  right  line  ;  and  therefore  the  bodies 
must  equally  press  the  obstacle,  and  be  equally  attracted  one  by 
the  other. 

"  I  made  the  experiment  on  the  loadstone  and  iron.  If  these  placed 
apart  in  proper  vessels,  are  made  to  float  by  one  another  in  standing 
water  ;  neither  of  them  will  propel  the  other,  but  by  being  equally 
attracted,  they  will  sustain  each  others  pressure,  and  rest  at  last  in 
equilibrium.  " 

4.  NEWTON  AND  THE  DYNAMICAL  LAW  OF  COMPOSITION  OF  FORCES. 

We  have  seen  how  Stevin  and  Roberval  had  established  the  rule  of 
the  composition  of  forces  in  statics.  Newton  arrived  at  the  law  of  the 
parallelogram  of  forces  by  purely  dynamical  considerations. 


208  THE   FORMATION  OF   CLASSICAL  MECHANICS 

"  Corollary  I  (to  the  second  law).  —  A  body  by  two  forces  conjoined 
will  describe  the  diagonal  of  a  parallelogram,  in  the  same  time  that  it  would 
describe  the  sides,  by  those  forces  apart. 

"  If  a  body  in  a  given  time,  by  the  force 
M  impressed  apart  in  the  place  A,  should 
with  an  uniform  motion  be  carried  from  A 
to  B  ;  and  by  the  force  N  impressed  apart 
in  the  same  place,  should  be  carried  from 
A  to  C —  compleat  the  parallelogram 
76  ABCD,  and  by  both  forces  acting  together, 

it  will  in  the  same  time  be  carried  in  the 

diagonal  from  A  to  D.  For  since  the  force  N  acts  in  the  direction  of 
the  line  AC,  parallel  to  BD,  this  force  (by  the  second  law)  will  not  at 
all  alter  the  velocity  generated  by  the  other  force  M,  by  which  the  body 
is  carried  towards  the  line  BD.  The  body  therefore  will  arrive  at  the 
line  BD  in  the  same  time,  whether  the  force  N  be  impressed  or  not  ; 
and  therefore  at  the  end  of  that  time,  it  will  be  found  somewhere  in  the 
line  jBD.  By  the  same  argument,  at  the  end  of  the  same  time  it  will 
be  found  somewhere  in  the  line  CD.  Therefore  it  will  be  found  in  the 
point  D,  where  both  lines  meet.  But  it  will  move  in  a  right  line  from  A 
to  D  by  Law  I.  " 

Newton's  demonstration  is  clearly  based  on  the  postulate  of  the 
independence  of  forces.  The  words  "  with  an  uniform  motion  "  show 
that  he  considered  the  impulsion  of  the  force  M  or  N  during  an  infinitely 
short  time.  This  force  acts  instantaneously,  like  a  percussion — this 
explains  the  importance  of  the  laws  of  impact  in  Newton's  thought. 
The  students  of  mechanics  in  the  XVIIth  Century  had  all  perceived  that 
the  phenomenon  of  impact  was  a  means  of  crystallising  the  effect  of  a 
force  into  the  velocity  acquired  in  a  first  instant. 

"  Corollary  II.  —  And  hence  is  explained  the  composition  of  any  one 
direct  force  AD,  out  of  any  two  oblique  forces  AB  and  BD  ;  and,  on  the 
contrary  the  resolution  of  any  one  direct  force  AD  into  two  oblique  forces 
AB  and  BD —  which  composition  and  resolution  are  abundantly  confirmed 
from  Mechanics.  " 

Newton  deduces  the  condition  of  equilibrium  for  simple  machines 
(the  balance,  inclined  plane  and  wedge)  from  this  proposition. 

We  have  already  seen  that  Aristotle  compounded  motions  according 
to  the  rule  of  the  parallelogram.1  Since  the  force  was  the  determinant 
of  the  velocity  in  his  belief,  it  may  be  held,  as  Duhem  has  done,  that 
Aristotle  compounded  forces  in  the  same  way. 

1  See  above,  p.  21. 


NEWTON  209 

For  Newton  too,  the  composition  of  forces  according  to  the  rule  of 
the  parallelogram  had  an  origin  in  dynamics.  But,  to  him,  the  force 
was  the  generator  of  a  quantity  of  motion  in  a  given  elementary  time 
(Definition  VIII,  para.  2  above). 

5.  THE  MOTION  OF  A  POINT  UNDER  THE  ACTION  OF  A  CENTRAL  FORCE. 

By  a  very  simple  and  direct  geometrical  argument,  Newton  esta 
blished  that  the  motion  of  a  material  point  that  is  subject  to  a  central 
force  was  contained  in  a  plane,  and  followed  the  law  of  areas  which 
Kepler  had  formulated  in  a  semi-empirical  way  (the  radius  vector 
sweeps  through  equal  areas  in  equal  times).  Here  is  Newton's  argument. 

"  Suppose  the  time  to  be  divided  into  equal  parts,  and  in  the  first 
part  of  that  time,  let  the  body  by  its  innate  force  describe  the  right 
line  AB.  In  the  second  part  of  that  time,  the  same  would  (by  Law  I), 
if  not  hindered,  proceed  directly  to  c,  along  the  line  Be  equal  to  AB  ; 
so  that  by  the  radii  AS,  BS,  cS 
drawn  to  the  centre,  the  equal 
areas  A  SB,  J3Sc,  would  be  de 
scribed.  But  when  the  body  arriv 
ed  at  B,  suppose  that  a  centripetal 
force  acts  at  once  with  a  great  im 
pulse,  and  turning  aside  the  body 
from  the  right  line  Be,  compells  it 
afterwards  to  continue  its  motion 
along  the  right  line  J5C.  Draw  cC 
parallel  to  BS  meeting  JBC  in  C  ; 
and  at  the  second  part  of  the  time, 
the  body  (by  Cor.  I  of  the  laws)  will  be  found  in  C,  in  the  same  plane 
with  the  triangle  ASB*  Join  SC,  and,  because  SB  and  Cc  are  parallel, 
the  triangle  SBC  will  be  equal  to  the  triangle  SJ3c,  and  therefore  also 
to  the  triangle  SAB. 

"  By  the  like  argument,  if  the  centripetal  force  acts  successively 
in  C,  D,  E,  and  c  and  makes  the  body  in  each  single  particle  of  time,  to 
describe  the  right  lines  CD,  DE,  EF,  and  c  they  will  all  lye  in  the  same 
plane  ;  and  the  triangle  SCD  will  be  equal  to  the  triangle  SBC,  and 
SDE  to  SCD,  and  SEF  to  SDE.  And  therefore  in  equal  times,  equal 
areas  are  described  in  one  immoveable  plane —  and,  by  composition, 
any  sums  SADS,  SAFS,  of  those  areas,  are  one  to  the  other,  as  the  times 

1  We  shall  encounter  this  method  of  argument  again  in  the  next  paragraph.    It  is 
equivalent  to  making  use  of  the  deviation  (in  the  kinematic  sense)  produced  by  the 
force  during  an  infinitely  short  time. 
14 


210  THE   FORMATION   OF   CLASSICAL  MECHANICS 

in  which  they  are  described.  Now  let  the  number  of  those  triangles  be 
augmented,  and  their  breadth  diminished  in  infinitum  ;  and  their  ulti 
mate  perimeter  ADF  will  be  a  curve  line " 

Newton  establishes  the  converse  of  this  proposition.  He  then  exa 
mines  the  circular  trajectory  of  a  body  gravitating  about  the  centre  of 
this  trajectory — its  gravity  is  equal  to  the  centripetal  force.  Therefore 
the  gravity  can  be  evaluated  by  using  the  propositions  given  by 
Huyghens  in  his  Horologium  oscittatorium.1 

Newton  then  studies  a  particle  which  describes  a  circular  orbit  under 
the  action  of  a  force  emanating  from  any  point  in  the  plane  of  the  circle. 
Given  this,  he  comes  to  the  fundamental  problem  of  the  motion  of  the 
planets. 

6.  NEWTON'S  EXPLANATION  OF  THE  MOTION  OF  THE  PLANETS. 

We  shall  quote  the  original  text  of  the  Principia  (De  motu  corporum. 
Liber  I9  Prop.  VI,  cor.  5). 

"  Si  corpus  P  revolvendo  circa  centrum  S  describat  lineam  curvam 
APQ  ;  tangat  vero  recta  ZPR  curvam  illam  in  puncto  quovis  P  et  ad 


Fig.  78 


tangentem  ab  alio  quovis  curvae  puncto  Q  agatur  QR  distantiae  SPparallela, 
ac  demitatur  QT  perpendicularis  ad  distantiam  illam  SP  :  vis  centripeta 

SPquad  X  QT  quad 

ent  reciproce  ut  sohdum  — si    modo  solidi  illius  ea 

QR 

semper  sumatur  quantitas  quae  ultimo  sit.,  ubi  coeunt  puncta  P  et  Q. 

44  Nam  QR  aequalis  est  sagittae  dupli  arcus  QP  in  cujus  media  est  P  ; 
et  duplum  trianguli  SPQ,  sive  SP  X  QT,  tempori  quo  arcus  iste  duplus 
describitur^proportionale  est ;  ideoquepro  temporis  exponente  scribi  potest." 

1  These  are  the  propositions  which  HUYGHENS  included,  -without  proof,  at  the  end 
of  his  treatise. 


NEWTON 


211 


That  is,  "  If  a  body  P  revolving  about  tke  centre  S,  describes  a 
curve  line  APQ,  which  a  right  line  ZPR  touches  in  any  point  P  ;  and 
from  any  other  point  Q  of  the  curve,  QR  is  drawn  parallel  to  the  distance 
SP,  meeting  the  tangent  in  R  ;  and  QT  is  drawn  perpendicular  to  the 
distance  SP —  the  centripetal  force  will  be  reciprocally  as  the  solid 


SP2  •  QT2 
QR 


-,  if  the  solid  be  taken  of  that  magnitude  which  it  ultimately 


acquires  when  the  points  P  and  Q  coincide. 

"  For  QR  is  equal  to  the  versed  sine  (sagitta)  of  double  the  arc  ()P, 
whose  middle  is  P —  and  double  the  triangle  SQP,  or  SP  X  QT  is  pro 
portional  to  the  time,  in  which  that  double  arc  is  described  ;  and  there 
fore  may  be  used  for  the  exponent  of  the  time.  " 

No  purpose  is  served  in  indicating  the  generality  and  the  remarkably 
direct  character  of  this  argument.  The  quantity  QR  is  now  called,  in 
kinematics,  the  deviation. 

*  dt2 
This  deviation  has  the  value  y  — ,  where  y  is  the  acceleration.   Now 

2 

since,  here,  the  acceleration  is  central,  like  the  force,  and  since  it  passes 
through  the  pole  S,  it  is  seen  that  QR  is  parallel  to  SP.  Since  the  law 
of  areas  is  applicable,  the  area  of  the  triangle  SPQ  is  proportional  to  dt. 
Since  the  force  is  itself  proportional  to  the  acceleration  it  is  therefore, 

SP2-QT2 

in  the  last  analysis,  inversely  proportional  to  the  expression . 

QR 


Fig.  79 


212  THE   FORMATION   OF   CLASSICAL   MECHANICS 

Newton  applies  this  general  law  to  trajectories  which  are  conic 
sections.  We  shall  confine  ourselves  here  to  the  elliptic  trajectory  and 
shall  summarise  the  solution  of  Problem  VI,  proposition  XI  —  "  Revol- 
vatur  corpus  in  ellipsi  :  requiritur  lex  vis  centripetae  tendentis  ad  umbi- 
licum  ellipseos.  " 

The  original  solution  has,  to  some  extent,  the  character  of  a  rebus  — 
we  shall  attempt  to  distinguish  the  essential  steps  and  to  express  them 
in  a  way  that  will  make  this  argument  clearer. 

If  DK  is  the  diameter  conjugate  to  CP,  Newton  first  verifies  that 
PE  =  a.  Then  drawing  the  line  Qxv  parallel  to  the  tangent,  cutting 

QR       PE         a 
SP  and  PC  in  x  and  v9  he  verifies  that  -=j—  =  -5—  =  -^  because  of  the 

JT  V  Jrd  Jr(^i 

similar  triangles.     In  the  same  way,  if  QT  is  perpendicular  to  SP  and 
PF  is  perpendicular  to  the  tangent, 

Qx  _    a 


Further,  by  Apollonius'  Theorem 

_o_       CD 
PF~~    b  " 
Hence 


Now  in  the  limit  when  Q  tends  to  P,  —  tends  to  unity.      Therefore 

Qx 


We  form  the  expression 


,.  -  ..      ^2 

hm  —  pr—  —  =  hm  SP2-^= 
QR 


By  the  ecpiation  of  the  ellipse  referred  to  oblique  conjugate  axes 
CD  and  CP 


CD2       CP2 
whence 


Qv 2  =  CP2  -  Cv2  _  (CP+Cv)Pv 
Cl)2  ~~        CP2        ~          CP2 


NEWTON  213 

Therefore 

limJg^^lim;^^ 

QR  CP*-a-Pv  aPC*  * 

Whence  the  conclusion — 

2J2     

"  Vis  centripeta  reciproce  e$t  ut  —  •  SP2,   id  est  reciproce   in  ratione 

a 

duplicata  distantiae  SP.  "     The  law  of  force  is  inversely  proportional  to 
the  distance. 

In  short,  this  proof  rests  on  the  newtonian  definition  of  force  ;  on 
the  use  of  the  kinematic  idea  of  deviation  ;  and  on  a  direct  argument  of 
infinitesimal  geometry  making  use  of  the  classical  properties  of  conies. 
Except  for  the  finite  properties  of  conies,  all  its  steps  were  unknown  to 
Newton's  predecessors,  and  were  indispensible  for  the  justification  of 
Kepler*  semi-empirical  laws  and  for  the  fashioning  of  celestial  mechanics 
into  a  chapter  of  dynamics. 

7.    THE    UNIVERSAL  ATTRACTION. 

The  scope  of  this  work  does  not  allow  us  to  deal  with  the  numerous 
problems  that  are  treated  in  the  Principia.  We  shall  only  describe  the 
path  that  was  travelled  by  Newton's  predecessors,  and  by  Newton 
himself,  and  which  ended  in  the  law  of  universal  attraction.1 

Only  an  excessive  schematisation  can  make  the  spontaneous  blossom 
ing  of  a  physical  theory  credible.  The  fall  of  an  apple  did  not  suffice 
to  give  Newton  the  idea  of  universal  gravitation — rather,  this  was  the 
product  of  a  long  development. 

As  Early  as  the  Xlllth  Century  Pierre  de  Maricourt,  in  a  letter 
written  in  castris  in  1269,  analysed  the  polarities  of  a  magnet  in  a  very 
detailed  way — the  magnetic  property  tends  to  conserve  the  integrity 
of  the  magnet  by  binding  its  parts  together. 

We  have  seen  how  the  Schools  of  the  XlVth  Century,  in  the  persons 
of  Jean  de  Jandun,  William  of  Ockham  and  Albert  of  Saxony,  discussed 
the  possibility  of  action  at  a  distance.2  We  have  seen  how  Copernicus 
maintained  that  gravity  was  only  a  "  natural  desire  "  given  to  the  parts 
of  the  Earth  in  order  that  their  integrity  might  result.3 

In  De  sympathia  et  antipathia  rerum  (1555),  Frascator  held  that  when 
two  parts  of  the  same  whole  were  separated  from  each  other,  each  of 


1  For  farther  details  the  reader  should  consult  DUHEM'S  Tkeorie  physique  (Paris, 
">),  pp.  364  et  se 

2  See  above,  p 

3  Ibid.,  p.  85. 


1906),  pp.  364  et  seq. 

2  See  above,  pp.  57-58. 


214  THE   FORMATION   OF  CLASSICAL  MECHANICS 

them  emitted  a  species  which  was  propagated  in  the  intermediate  space. 

In  De  Magnete  (London,  1600),  Gilbert  argued  that  the  rectilinear 
motion  of  heavy  bodies  was  the  motion  of  the  reunion  of  separated 
parts.  He  added  that  "  this  motion,  which  is  only  the  inclination 
towards  its  source,  does  not  only  belong  to  the  parts  of  the  Earth,  but 
also  to  the  parts  of  the  Sun,  the  Moon,  and  to  those  of  the  other  celestial 
orbs.  "  Here  Gilbert  enters  on  the  metaphysical  plane.  "  We  give 
the  cause  of  this  coming  together  and  this  motion  which  touches  all 
nature.  ...  It  is  a  substantial  form  which  is  special,  particular,  belong 
ing  to  primary  and  principal  spheres  ;  it  is  a  proper  entity  and  an  essence 
of  their  homogeneous  and  their  uncorrupted  parts,  which  we  call  a  pri 
mary,  radical  and  astral  form ;  it  is  not  Aristotles'  first  form,  but  that 
special  form  by  which  the  orb  conserves  and  disposes  what  is  its  own.  .  .  . 
It  constitutes  that  true  magnetic  form  that  we  call  the  primary  energy. " 1 

This  animist  philosophy,  as  Duhem  has  called  it,  was  adopted  by 
Francis  Bacon.  Kepler  himself  was  stimulated  by  it,  but  substituted 
in  it  the  idea  on  one  single  property  belonging  to  any  part  of  any  star. 
We  have  already  remarked  2  that  Kepler  regarded  gravity  as  a  "  mutual 
affection  between  parent  bodies  that  tends  to  unite  them.  " 

As  far  as  the  tides  are  concerned,  Ptolemy  had  already  produced 
an  explanation  by  invoking  a  special  influence  of  the  Moon  on  the  seas. 
In  order  to  get  rid  of  what  seemed  to  them  an  occult  quality,  Averroes, 
Albertus  Magnus  and  Roger  Bacon  attributed  this  action  to  the  heat 
of  the  light  from  the  Moon.  Albert  of  Saxony  championed  an  animist 
theory  of  the  tides.  Cardan,  followed  by  Scaliger,  believed  only  in  an 
obedience  of  the  waters  to  the  Moon. 

Kepler  himself  wrote,  "  Observation  proves  that  everything  that 
contains  humidity  swells  when  the  Moon  waxes  and  shrinks  when  the 
Moon  wanes.  "  3  But  later  he  corrected  this  opinion,  and  thereby  anti 
cipated  the  Newtonian  thesis. 

"  The  Moon  acts  not  as  a  moist  or  damp  star,  but  as  a  mass  similar 
to  the  mass  of  the  Earth.  It  attracts  the  waters  of  the  sea,  not  because 
they  are  fluids  but  because  they  are  gifted  with  terrestrial  substance,  to 
which  they  also  owe  their  gravity.  "  4 

This  attraction  is  reciprocal.  "  If  the  Moon  and  the  Earth  were  in 
no  way  held  by  a  sensual  force  or  by  some  equivalent  force,  each  in  its 
orbit,  the  Earth  would  rise  towards  the  Moon  and  the  Moon  would 
descend  towards  the  Earth  until  these  two  stars  joined  together.  If  the 

1  Translated  into  French  by  DUHEM. 

2  See  above,  p.  118. 

3  Opera  omnia,  Vol.  I,  p.  422. 

4  Ibid.,  Vol.  VII,  p.  118. 


NEWTON  215 

Earth  ceases  to  attract  the  waters  that  cover  it  to  itself,  the  waves  of 
the  sea  would  all  rise  and  run  towards  the  body  of  the  Moon.  "  * 

Returning  to  a  thesis  which  had  already  been  put  forward  by 
Calcagnini,  Galileo  held  that  the  ebb  and  the  flow  of  the  sea  was  explain 
ed  by  the  following  relative  motion.  The  Earth  turns  from  East  to 
West  at  the  same  time  that  it  is  animated  by  a  translational  velocity  v. 
At  a  the  two  motions  add  together — at  6, 
they  tend  to  cancel  out.  Because  of  their 
inertia,  the  waters  of  the  sea  do  not  follow 
this  motion  exactly.  The  ebb  and  flow, 
thanks  to  the  delay,  is  produced  twice  a 
day  although,  if  the  composition  of  the 
motions  were  perfect,  they  would  have  the 
period  of  the  rotation  of  the  Earth. 
Therefore  Galileo  interpreted  the  tidal 
phenomena  as  proof  of  the  motion  of  the 
Earth,  while  the  opponents  of  the  copernican  system  held  to  a  lunar 
attraction. 

The  astrologers  of  the  XVIth  Century,  following  Grisogone,  were 
inspired  to  separate  the  whole  tide  into  a  solar  tide  and  a  lunar  tide. 
In  1528  Grisogone  wrote,  "  The  Sun  and  the  Moon  attract  the  swelling 
of  the  sea  towards  themselves,  so  that  the  maximum  swelling  is  perpen 
dicularly  beneath  each  of  them.  Therefore,  for  each  of  them,  there  are 
two  maxima  of  swelling,  one  beneath  the  star  and  one  on  the  opposite 
side,  which  is  called  the  nadir  of  the  star.  " 

Ideas  on  the  law  of  attraction  itself  were  yet  more  vague  and  chan 
geable.  To  Roger  Bacon,  all  actions  at  a  distance  were  propagated  in 
straight  rays,  like  light.  Kepler  took  up  this  analogy — now,  it  has  been 
known  since  the  time  of  Euclid  that  the  intensity  of  the  light  emitted 
by  a  source  varies  in  inverse  ratio  of  the  square  of  the  distance  from  the 
source.  In  this  optical  analogy,  the  virtus  movens  emanating  from  the 
Sun  and  acting  on  the  planets  must  follow  the  same  law.  But  in  dyna 
mics  Kepler  remained  an  Aristotelian — force  was,  to  him,  proportional 
to  velocity.  Therefore  Kepler  deduced  the  following  result  from  the 
law  of  areas  rv  =  constant.  The  virtus  movens  of  the  Sun  on  the  planets 
is  inversely  proportional  to  the  distance  from  the  Sun.  In  order  to 
reconcile  this  law  with  the  optical  analogy  Kepler  held  that  light  spread 
out  in  all  directions  in  space,  while  the  virtus  movens  was  only  effective 
in  the  plane  of  the  solar  equator. 

Boulliau,  writing  Astronomia  Philolaica  in  1645,  carried  the  optical 

1  Opera  omnia,  Vol.  Ill,  p.  151. 


216  THE   FORMATION   OF   CLASSICAL   MECHANICS 

analogy  to  its  limit  and  supported  the  law  of  attraction  inversely  pro 
portional  to  the  square  of  the  distance.  But  it  should  be  remarked 
that  this  attraction  was  normal  to  the  radius  vector,  and  not  central 
as  the  Newtonian  theory  demanded. 

Descartes  confined  himself  to  replacing  Kepler's  virtus  movens  by 
a  vortical  ether. 

BoreUi  has  the  merit  of  having  invoked  the  example  of  the  sling  in 
order  to  explain  why  the  planets  did  not  fall  on  the  Sun— he  sets  the 
instinct  by  which  the  planet  carries  itself  towards  the  Sun  against  the 
tendency  of  aU  bodies  in  rotation  to  move  away  from  their  centre— this 
vis  repellens  is  inversely  proportional  to  the  radius  of  the  orbit. 

In  a  paper  caUed  An  Attempt  to  prove  the  annual  Motion  of  the  Earth 
(1674)  Hooke,  curator  of  the  Royal  Society,  clearly  formulated  the  prin 
ciple  of  universal  gravitation.  "  AU  celestial  bodies  without  exception 
exert  a  power  of  attraction  or  heaviness  which  is  directed  towards  their 
centre  ;  in  virtue  of  which  they  not  only  retain  their  own  parts  and 
prevent  them  from  escaping,  as  we  see  to  be  the  case  on  the  Earth,  but 
also  they  attract  all  the  celestial  bodies  that  happen  to  be  within  the 
sphere  of  their  activity.  Whence,  for  example,  not  only  do  the  Sun  and 
the  Moon  act  on  the  progress  and  motion  of  the  Earth  in  the  same  way 
that  the  Earth  acts  on  them,  but  also  Mercury,  Venus,  Mars,  Jupiter 
and  Saturn  have,  because  of  their  attractive  power,  a  considerable 
influence  on  the  motion  of  the  Earth  in  the  same  way  that  the  Earth 
has  an  influence  on  the  motion  of  these  bodies.  "  Hooke  assumed  that 
the  attraction  decreased  with  the  distance  and,  in  1672,  declared  him 
self  for  the  inverse  square  law.  No  doubt  he  was  guided  by  the  optical 
analogy. 

In  order  to  justify  this  result  it  was  necessary  to  know  the  laws  of 
centrifugal  force.  Now  we  know  that  although  Huyghens  had  written 
his  treatise  De  vi  centrifuga  as  early  as  1659,  only  the  statements  of  the 
thirteen  propositions  that  conclude  the  Horologium  oscillatorium  were 
published  during  his  lifetime. 

Halley  appears  to  have  applied  Huyghens'  theorems  to  Hooke's 


hypothesis.  By  assuming  Kepler's  third  law  (—  =  constant)  he  dis 
covered  the  law  of  the  inverse  square. 

This  whole  development,  that  we  have  only  been  able  to  summarise, 
shows  that  one  cannot  talk  of  the  spontaneous  generation  of  the  theory 
of  gravitation. 

For  his  part,  Newton  was  in  possession  of  the  laws  of  uniform  circular 
motion  in  1666.  By  an  analysis  analogous  to  that  which  Halley  had 
made,  and  starting  from  Kepler's  third  law,  he  formulated  the  law  of 


NEWTON  217 

an  attraction  inversely  proportional  to  the  square  of  the  distance.  But 
more  careful  than  his  predecessors,  Newton  sought  experimental  veri 
fication  for  this  law.  He  tried  to  discover  whether  the  attraction  exert 
ed  by  the  Earth  on  the  Moon  corresponded  to  this  law,  and  whether  this 
attraction  could  be  identified  with  terrestrial  heaviness. 

Since  the  radius  of  the  Earth's  orbit  is  of  the  order  of  60  terrestrial 
radii,  the  force  that  maintains  the  Moon  in  its  orbit  is  3600  times  weaker 
than  the  heaviness  at  the  centre  of  the  Earth.  Now  a  body  falling 
freely  in  the  neighbourhood  of  the  Earth  falls  a  distance  of  15  Paris 
feet  l  in  the  first  second.  The  Moon  would  therefore  fall  a  distance  of 

—  pouce  in  the  first  second.     Knowing  the  period  of  the  Moon's  motion 

and  the  radius  of  its  orbit,  it  is  easy  to  calculate  this  fall  of  the  Moon, 
With  the  data  on  the  Earth's  radius  that  were  accepted  in  England, 

Newton  obtained  a  fall  of  only  —  pouce. 

2>& 

Faced  with  this  divergence,  he  gave  up  his  idea.  It  was  only  16 
years  later  (1682)  that  he  learnt  of  the  measurement  of  the  terrestrial 
meridian  that  had  been  made  by  Picard.  (This  happened  at  a  meeting 
of  the  Royal  Society.)  By  assuming  the  value  given  by  this  determin 
ation,  Newton  obtained  the  expected  value  of  —  pouce.  He  was  then 

250 

able  to  declare,  "  Lunam  gravitare  in  Terrain  et  vi  gravitatis  retrahi 
semper  a  motu  rectilineo  et  in  orbe  suo  retineri  " ;  and,  by  an  induction 
conforming  to  the  very  principles  of  his  philosophy,  to  affirm  the  doctrine 
of  universal  gravitation. 

The  theory  of  the  attraction  of  spheres  allowed  him  to  concentrate 
at  their  centres  the  masses  of  stars  that  were  supposed  to  be  formed 
of  homogeneous  concentric  layers,  and  thus  to  reduce  them  to  material 
points  whose  mutual  attractions  could  be  studied. 

Newton  evaluated  the  masses  and  densities  of  the  Sun  and  the  planets 
that  were  surrounded  by  satellites.  He  also  calculated  the  heaviness 
at  a  point  on  their  surface.  He  showed  that  the  comets  described 
very  elongated  elliptical  trajectories  and  replaced  these  by  parabolas 
whose  elements  he  calculated.  In  this  way  he  was  able  to  connect  the 
segments  of  trajectory  of  a  comet  that  had  appeared  on  each  side  of 
the  Sun  in  1680.  Halley  then  showed  that  the  appearances  in  1531, 
1607  and  1682  were  those  of  this  same  comet. 

Newton  also  showed  that  the  rotation  of  the  Earth  must  entail 
its  flattening  at  the  two  poles,  and  calculated  the  variation  of  gravity 

1  In  these  discussions  in  the  Priracipia,  the  distances  are  given  in  French  units. 


218  THE  FORMATION   OF   CLASSICAL  MECHANICS 

along  a  meridian.  He  related  the  theory  of  tides  to  the  combined 
attraction  of  the  Moon  and  the  Sun  and  thus  justified  the  anticipations 
of  the  astrologers  of  the  XVIth  Century.  Finally,  calculating  the 
actions  of  the  Moon  and  the  Sun  on  the  equatorial  bulge,  he  arrived  at 
a  theory  of  the  precession  of  the  equinoxes. 


CHAPTER   SEVEN 


LEIBNIZ  AND  LIVING  FORCE 


1.  THE  "  vis  MOTRIX  "  m  THE  SENSE  OF  LEIBNIZ. 

Leibniz  protested  against  the  cartesian  mechanics  in  a  memoir 
which  appeared  in  1686  in  the  Ada  eruditorum  at  Leipzig,  under  the 
title  A  short  demonstration  of  a  famous  error  of  Descartes  and  other 
learned  men,  concerning  the  claimed  natural  law  according  to  which 
God  always  preserves  the  same  quantity  of  motion ;  a  law  which  they 
use  incorrectly ,  even  in  mechanics. 

Leibniz  set  out  to  show  that  the  vis  motrix  (or,  in  the  words  of  the 
XVIIIth  Century,  the  force  of  bodies  in  motion),  was  distinct  from 
the  quantity  of  motion  in  Descartes'  sense. 

Like  Huyghens,  Leibniz  assumes  that  a  body          < 
falling  freely  from  a  given  height  will  acquire 
the  "  force  "  necessary  to  rise  again  to  the  same 
height,    if   the    resistance    of    the    medium    is 
neglected  and   no   external  inelastic  obstacle  is  ~ 

encountered.      On   the    other    hand,    like   Des-  r 

cartes,  he  assumes  that  the  same  "  force  "   (in  ^-^ 

the  modern  sense  of  work)  is  needed  to  lift  a 
body  A,  whose  weight  is  one  pound,  to  a  height  -p.     ^ 

DC  of  four  ells  as  to  lift  a  body  J3,  whose  weight 
is  four  pounds,  to  a  height  of  one  ell. 

In  falling  freely  from  the  height  CD  the  body  A  acquires  the  same 
"  force  "  as  the  body  B  acquires  in  falling  from  the  height  EF.  For 
when  it  has  arrived  at  D,  the  body  A  has  acquired  the  force  that  it 
needs  to  climb  again  to  C,  and  the  body  J5,  when  it  has  come  to  F, 
has  acquired  the  force  needed  to  climb  to  E.  By  hypothesis,  these 
two  forces  are  equal.  Now  the  quantities  of  motion  of  A  and  B  are 
far  from  being  equal. 

Indeed,  Galileo's  laws  show  that  the  velocity  acquired  in  the  free  fall 
CD  is  twice  the  velocity  acquired  in  the  free  fall  EF.  The  quantity 


220  THE   FORMATION   OF   CLASSICAL   MECHANICS 

of  motion  of  A  is  then  proportional  to  1x2,  while  that  of  B  is  pro 
portional  to  4x1,  and  is  therefore  twice  that  of  A.  This  contradicts 
the  cartesian  thesis  in  which  the  quantity  of  motion  is  used  to  evaluate 
the  "  force.  " 

Leibniz  recognised  that  in  simple  machines  (the  lever,  the  windlass, 
the  pulley,  the  wedge  and  the  screw)  the  same  quantity  of  motion 
tended  to  be  produced,  in  one  part  and  the  other,  when  equilibrium 
obtained.  "  Thus  it  happens  by  accident  that  the  force  can  be  reckoned 
as  the  quantity  of  motion.  But  there  are  other  instances  in  which 
this  coincidence  no  longer  exists.  " 

And  Leibniz  concludes,  "  It  should  be  said,  therefore,  that  the 
forces  are  in  compound  proportion  to  the  bodies  (of  the  same  specific 
weight  or  density)  and  the  generating  heights  of  the  velocities — that 
is,  the  heights  from  which  the  bodies  are  able  to  acquire  their  velo 
cities  in  falling,  or  more  generally  (since  often  no  velocity  has  been 
produced  at  this  point),  the  heights  that  will  be  generated.  "  x 

2.  LEIBNIZ  AND  THE  LAWS  OF  IMPACT. 

Writing  to  the  Abbe  de  Conti  in  1687,  Leibniz  suggested  that  for 
the  cartesian  principle  of  the  conservation  of  the  quantity  of  motion 
should  be  substituted  a  natural  law  which  he  took  as  universal  and 
inviolate.  This  was,  "  that  there  is  always  a  perfect  equality  between 
the  complete  cause  and  the  whole  effect.  " 

In  this  connection  he  went  on  to  discuss  Descartes'  third  rule 
on  the  impact  of  bodies.2 

"  Suppose  that  two  bodies,  B  and  C,  each  weighing  one  pound 
and  travelling  in  the  same  direction,  collide  with  each  other.  The 
velocity  of  B  is  100  units  and  that  of  C,  1  unit.  Their  total  quantity 
of  motion  will  be  101.  But  if  C,  with  its  velocity,  can  rise  to  a  height 
of  one  pouce,  the  velocity  of  B  will  enable  it  to  rise  to  a  height  of  10,000 
pouces.  Thus  the  force  of  the  two  united  bodies  will  be  that  of  lifting 
one  pound  to  10,001  pouces.  Now  according  to  Descartes  third  rule, 
after  the  impact  the  bodies  will  go  together  in  company  with  a  common 
velocity  of  50  and  a  half.  .  .  .  But  then  these  2  pounds  are  only  able 
to  lift  themselves  to  a  height  of  2550  pouces  and  a  quarter,  which 
is  equivalent  to  lifting  one  pound  to  5100  pouces  and  a  half.  Thus 
almost  half  the  force  will  be  lost  according  to  this  rule,  without  there 
being  any  reason  and  without  its  having  been  used  for  anything.  " 

1  Translated  into  French  by  JOUGUET. 

2  See  above,  p.  162. 


LEIBNIZ 


221 


In  this  discussion  lies  the  germ  of  the  controversy  about  living 
forces  that  was  to  divide  the  geometers  at  the  beginning  of  the  XVIIIth 
Century,  and  to  which  we  shall  return.  We  know  now  that  Descartes 
third  rule  is  correct  and  is  applicable  to  perfectly  soft  bodies  (soft, 
in  order  that  they  should  travel  together  after  the  impact).  The 
total  quantity  of  motion  is  conserved  (no  difficulty  of  sign  occurs 
here)  and  a  part  of  the  living  force  is  transformed  into  heat. 

3.   LIVING    AND    DEAD    FORCES. 

Leibniz  showed  himself  to  be  even  more  systematic  in  his  Specimen 
dynamicum  (1695). 

We  shall  pass  over  the  several  quantities  that  he  introduced  and 
only  concern  ourselves  with  the  distinction  between  living  forces  and 
dead  forces. 

44  Force  is  twin.  The  elementary  force,  which  I  call  dead  because 
motion  does  not  yet  exist  in  it,  but  only  a  solicitation  to  motion,  is 
like  that  of  a  sphere  in  a  rotating  tube  or  a  stone  in  a  sling. 

44  The  other  is  the  ordinary  force  associated  with  actual  motion, 
and  I  call  it  living. 

44  Examples  of  dead  force  are  provided  by  centrifugal  force,  by 
gravity  or  centripetal  force,  and  by  the  force  with  which  a  stretched 
spring  starts  to  contract. 

44  But  in  percussion  that  is  produced  by  a  body  which  has  been 
falling  for  some  time,  or  by  an  arc  which  has  been  unbending  for  some 
time,  or  by  any  other  means,  the  force  is  living  and  born  of  an  infinity 
of  continued  impressions  of  the  dead  force.  " 

Leibniz  reproached  the  Ancients  "  for  having  had  exclusively  an 
understanding  of  dead  forces,  and  for  only  having  studied  the  first 
conatus  [in  Huyghens  sense]  of  bodies  to  each  other,  even  though  the 
latter  had  not  acquired  an  impetus  [in  the  sense  of  quantity  of  motion] 
by  the  action  of  the  forces. 

In  modern  language,  Leibniz's  assertion  that  the  living  force  is 
born  of  an  infinity  of  impressions  of  the  dead  force  may  be  expressed  by 


This  leads  to  the  fundamental  law  m  —  =  F  and  identifies  the  dead 

at 

force  as  the  static  force. 


CHAPTER   EIGHT 


THE  FRENCH  -  ITALIAN  SCHOOL  OF  ZACCHI  AND  VARIGNON 


1.  ZACCHI  AND  SACCHERI.  LAMY  AND  THE  COMPOSITION  OF  FORCES. 

We  shall  devote  this  chapter  to  a  brief  analysis  of  some  works 
which  can  only  appear  as  miniatures  in  comparison  with  those  of 
Galileo,  Huyghens  and  Newton.  But  in  leaving  the  peaks  on  which 
the  work  of  the  creators  of  dynamics  lies,  we  shall  have  a  better  appre 
ciation  of  the  extent  to  which  those  dominated  their  own  century. 

In  a  Nova  de  machinis  philosophia  (Roma,  1649)  Zacchi  was  concern 
ed  with  an  attempt  to  isolate  the  principles  implicit  in  Aristotle's 
statics.  Under  the  term  virtus  he  confused  the  concepts  of  force 
and  work,  and  thus  misunderstood  Descartes'  principle. 

Father  Fabri  (1606-1688)  was  a  teacher  at  the  Jesuit  College  at 
Lyons  and  a  friend  of  Mersenne.  His  Tractatus  physicus  de  motu 
locali  (1646)  was  a  work  on  dynamics  which  took  over  the  ideas  of 
Jordanus  and  Albert  of  Saxony.  Among  the  moderns  it  only  makes 
mention  of  Galileo's  statics — and  this,  as  we  have  seen,  was  impregnated 
with  the  ideas  of  Aristotle. 

Father  Lamy  attacked  Descartes  in  his  Treatise  on  Mechanics  (1679)  and 
contested  Stevin's  argument  on  the  inclined  plane.  He  claimed  that  no 
thing  proved  that  the  lower  part  of  the  chain  of  balls  hung  symmetrically. 

We  now  know  that  this  criticism  is  not  justified  and  that  if  the 
number  of  balls  is  infinite  the  necklace  outlines  a  perfectly  symme 
trical  catenary  underneath  the  plane. 

In  order  to  solve  the  problem  of  the  equilibrium  of  a  body  on  an 
inclined  plane,  Lamy  preferred  to  return  to  the  arguments  of  Bernardino 
Baldi  and  da  Vinci. 

To  set  against  this  is  a  letter  addressed  by  Lamy  to  M.  de  Dieulamant, 
an  engineer  at  Grenoble,  which  is  concerned  with  the  law  of  the  com 
position  of  forces  and  deserves  a  little  of  our  attention. 

"  1.  When  two  forces  draw  the  body  Z  along  the  lines  AC  and  JBC, 
which  are  called  the  lines  of  direction  of  the  forces,  it  is  clear  that  the 


THE   FRENCH-ITALIAN   SCHOOL  223 

body  Z  will  not  travel  on  the  line  AC  or  on  the  line  JBC,  but  on  another 
line  between  AC  and  BC,  say  X. 

"  2.  If  the  path  X  were  closed  then  Z,  which  is  forced  to  travel 
by  this  path,  would  remain  motionless,  so  that  the  forces  would  be 
in  equilibrium. 


Fig.  82 

"  3.  Force  is  that  which  can  move  things.  Motions  are  only  mea 
sured  by  the  distances  which  they  travel.  Suppose  then  that  the  force 
A  is  to  the  force  B  as  6  is  to  2.  Then  if  A9  in  a  first  instant,  draws  the 
body  Z  as  far  as  the  point  E  on  its  own,  in  the  same  instant  B  would 

only  draw  it  as  far  as  F  I  CF  =  -  CE  ).     We  have  seen  that  Z  cannot 

\  3  ^    / 

go  along  AC  or  along  BC.  Thus  it  is  necessary  that  in  the  first  instant 
it  should  come  to  D,  where  it  corresponds  to  E  and  to  F — that  is  to  say, 
where  it  has  travelled  the  value  of  CE  and  of  FC.  .  .  . 

"  This  line  X  is  related  to  the  lines  of  direction  of  the  two  forces  A 
and  B  in  such  a  way  that  at  any  point  from  which  two  perpendiculars  on 
the  two  lines  are  drawn,  their  relation  to  each  other  will  be  the  reciprocal 
of  that  of  the  forces,  or  the  relation  of  DE  to  DF.  " 

Lamy's  demonstration  is  very  similar  to  that  on  Newton.  The 
simultaneity  (1687)  of  the  two  demonstrations  makes  it  seem  however, 
that  they  were  independent  of  each  other.  On  the  other  hand,  Lamy 
was  accused  of  plagiarism  from  Varignon,  who  published  his  plan  for 
a  new  mechanics  at  the  same  time.  Lamy  vigorously  defended  himself 


224  THE   FORMATION   OF    CLASSICAL   MECHANICS 

against  this  charge.  If,  like  Duhem,1  we  put  the  emphasis  on  the 
words  "  in  a  first  instant,  "  it  is  reasonable  to  believe  that  Lamy  used 
an  argument  which  would  have  been  acceptable  in  modern  mechanics. 
On  the  other  hand,  Varignon — who  only  cared  for  statics,  a  branch  of 
the  subject  in  which  he  showed  great  skill — did  not  progress  beyond 
Aristotle's  dynamics. 

We  must  also  say  a  little  about  Neostatique  (1703),  a  rather  original 
work  due  to  Father  Saccheri. 

Saccheri  regarded  the  vis  matrix  as  proportional  to  the  impetus,  the 
term  which  he  used  to  denote  the  absolute  value  of  the  velocity.  As  he 
was  not  concerned  with  the  impetus  of  a  body,  starting  from  rest,  in 
the  first  instant,  this  rule  is  equivalent  to  that  of  Aristotle.  However, 
Saccheri  arrived  at  an  accord  with  Newton's  dynamics.  Thus  he  called 
the  oriented  velocity  the  impetus  vivus,  and  used  the  term  impetus 
subnascens  for  a  quantity  which,  for  a  body  of  weight  p,  reduced  to  the 

projection  of  the  acceleration  —  on  the  tangent.     In  identifying  the 

m 

impetus  subnascens  as  the  incrementum  of  the  impetus  vivus,  he  was  able 
to  write  down  the  Newtonian  law  of  motion.  This  illustrates  the  extent 
to  which  the  language  and  the  ideas  of  the  XVIIth  Century  were 
confused. 

Father  Ceva  had  drawn  Saccheri's  attention  to  the  law  of  Beaugrand 
and  Fermat  which  we  have  mentioned  in  connection  with  the  contro 
versy  on  geostatics.2 

This  is  the  law  of  an  attraction  which  is  proportional  to  the  distance. 

Saccheri  had  the  merit  of  showing  that,  according  to  this  law,  the 
heaviness  passed  through  a  centre  of  gravity  that  was  fixed  in  the  body. 
Also,  that  a  body  falling  freely  from  rest  and  subject  to  this  law,  arrived 
at  the  common  centre  of  heavy  bodies  in  a  time  which  did  not  depend 
on  its  distance  from  the  centre. 


2.  THE  STATICS  OF  VARIGNON  (1654-1722). 

Varignon  produced  his  Project  for  a  New  Mechanics  in  1687,  and  the 
New  Mechanics  or  Statics  only  appeared  posthumously  in  1725.  At  the 
beginning  of  the  Project  Varignon  acknowledged  the  influence  of  Wallis 
and  that  of  Descartes.  The  latter  had  declared  that  it  was  "  a  ridicul 
ous  thing  to  wish  to  use  the  argument  of  the  lever  in  the  pulley  "  ; 
Varignon  persuaded  himself  that  it  was  equally  useless  to  treat  the 

1  0.  S.,  Vol.  II,  p.  259. 

2  See  above,  p.  166. 


THE    FRENCH-ITALIAN   SCHOOL  225 

inclined  plane  by  starting  from  the  lever.  Of  a  more  practical  mind 
than  his  predecessors,  he  attached  more  weight  to  a  study  of  the  modes 
of  equilibrium  than  to  its  necessity,  and  reduced  everything  to  the  prin 
ciple  of  compound  motions.  "  It  seems  to  me  that  the  physical  reason 
for  the  effects  that  are  most  admired  in  machines  is  exactly  that  of 
compound  motion.  " 

It  is  important  to  remark  that  Varignon  interpreted  the  composition 
of  forces  and  of  motions  in  Aristotle's  sense,  for  he  remained  consciously 
faithful  to  aristotelian  dynamics.  Indeed,  in  his  New  Mechanics  he 
wrote — 

"  Axiom  VI.  —  The  velocities  of  a  single  body  or  of  bodies  of  equal 
mass  are  as  all  the  motive  forces  which  are  there  used,  or  which  cause 
these  velocities  ;  conversely,  when  the  velocities  are  in  this  ratio  they 
are  those  of  a  single  body  or  of  bodies  of  equal  masses.  " 

To  Varignon,  all  force  is  analogous  to  the  tension  of  a  thread.  In 
the  diagrams  which  appear  in  his  books,  the  hands  holding  the  threads 
materialise  the  powers.  He  neglects  all  friction  and  even  heaviness, 
which  he  identifies  with  a  tension. 

"  Requirement  II.  —  That  it  may  be  permissible  to  neglect  the 
heaviness  of  a  body  and  to  consider  it  as  if  it  had  none ;  but  to  regard 
it  as  a  power  which  may  be  applied  to  the  weight ;  when  it  will  be  con 
sidered  as  weight,  notice  will  be  given.  ..." 

Varignon  starts  from  a  general  principle  which  he  expresses  in  the 
following  way. 

"  Whatever  may  be  the  number  of  forces  or  powers,  directed  as  may 
be  chosen,  that  act  at  once  on  the  same  body,  either  this  body  will  not 
be  displaced  at  all  ;  or  it  will  travel  along  one  path  and  along  a  line 
which  will  be  the  same  as  if,  instead  of  being  pushed  in  this  way,  com 
pressed  or  drawn  by  all  these  powers  at  once,  the  body  was  only  follow 
ing  the  same  line  in  the  same  direction  by  means  of  a  single  force  or 
power  equivalent  or  equal  to  the  resultant  of  the  meeting  of  all  those 
forces.  " 

Therefore  everything  reduces  to  the  determination  of  this  resultant. 
And  it  is  here  that  Varignon  affirms  his  allegiance  to  the  Ancients. 

"  It  is  what  we  are  going  to  find  by  means  of  compound  motions 
known  to  the  ancients  and  the  moderns —  Aristotle  treats  them  in  the 
problems  of  mechanics  ;  Archimedes,  Nicodemus,  Dinostratus,  Diocles, 
etc.  .  .  .  have  used  them  for  the  description  of  the  spiral,  the  conchoid, 
the  cissoid,  etc.  .  .  . ;  Descartes  used  them  to  explain  the  reflection  and 
refraction  of  light  ;  in  one  word,  all  mathematicians  use  compound 
motions  for  the  generation  of  an  infinity  of  curved  lines,  and  all  correct 
physicists  for  determining  the  forces  of  impact  or  of  oblique  percussions, 


226  THE   FORMATION   OF   CLASSICAL   MECHANICS 

etc.  . . .  Thus  I  claim  nothing  but  the  principle  I  indicated  nearly  forty 
years  ago,  and  that  I  use  once  more  for  the  explanation  of  machines.  " 

Given  this,  it  is  easy  to  see  how  Varignon  reduced  the  composition 
of  forces  to  that  of  velocities.  The  superiority  of  Varignon' s  work  in 
statics  is  a  didactic  one.  He  treats  all  simple  machines  in  detail  by 
means  of  the  composition  of  forces  alone — this  by  ingenious  procedures 
that  are  still  commonly  used. 

In  Duhem's  opinion,  it  does  not  seem  that  the  geometers  of  the 
XVIIth  Century,  and  even  of  the  XVIIIth  Century,  had  attached  any 
importance  to  the  distinction  that  can  now  be  made  between  the  method 
of  Newton  and  of  Lamy  on  the  one  hand,  and  of  Varignon  on  the  other, 
in  the  matter  of  the  proof  of  the  rule  of  the  parallelogram.  "  The  pro 
positions  that  aristotelian  dynamics,  over  a  period  of  two  thousand 
years,  had  made  customary  in  physics  were  also  familar  to  all  minds. 
They  were  still  invoked  naturally  on  all  occasions  when  conscience  did 
not  too  violently  conflict  with  the  truths  of  the  new  Dynamics.  When 
Varignon,  in  1687,  produced  his  Project  of  a  New  Mechanics,  he  took  as 
his  starting  point  axioms  which  were  said  to  have  been  borrowed 
from  Physica  auscultatio  or  De  Caelo ;  but  at  the  same  time  Newton  and 
Lamy  showed  that  the  same  consequences  could  be  obtained  from  an 
accurate  dynamics.  "  * 

3.  VARIGNON  AND  TORRICELLI'S  LAW  OF  FLOW. 

In  the  second  Book  of  the  Principia  Newton  had  undertaken  a  proof 
of  Torricelli's  law  of  flow.  He  remarked  that  a  column  of  liquid  falling 
freely  in  a  vacuum  assumed  the  shape  of  a  solid  of  revolution  whose 
meridian  was  a  curve  of  the  fourth  degree.  Indeed,  the  velocity  of 
each  horizontal  slice  is  proportional  to  the  square  root  of  the  height 
from  which  it  has  fallen.  On  the  other  hand,  this  same  velocity  is 
inversely  proportional  to  the  section  of  the  column,  and  consequently 
to  the  square  of  the  radius.  In  a  vessel  having  this  shape  and  kept 
filled  with  water,  it  is  clear  that  each  particle  of  the  fluid  has  its  velocity 
of  free  fall  and  that,  in  consequence,  Torricelli's  law  is  justified. 

Newton  then  imagines  that  in  a  cylindrical  vessel  whose  base  is 
pierced  with  a  hole,  the  fluid  separates  into  two  parts.  One,  the  cata 
ract,  takes  the  shape  of  free  fall  of  which  we  have  spoken.  The  other 
remains  motionless.  It  is  easy  to  see  that  this  solution  contradicts  the 
principles  of  hydrostatics. 

Varignon  had  the  merit  of  giving  TorricellFs  law  a  more  natural 

1  0.  S.,  Vol.  II,  p.  260. 


THE    FRENCH-ITALIAN    SCHOOL  227 

explanation.  He  assumed  that  the  water  remained  sensibly  immobile 
up  to  the  immediate  neighbourhood  of  the  hole.  At  that  point  each 
particle  instantaneously  received,  in  the  form  of  a  finite  velocity,  the 
effect  of  the  weight  of  the  fluid  that  was  above  it.  It  is  easy  to  see, 
taking  account  of  the  quantity  of  water  flowing  out,  that  the  quantity 
of  motion  thus  created  in  each  particle  is  proportional  to  the  square 
of  the  velocity.  If  the  weight  of  the  column  of  water  above  it  is  pro 
portional  to  the  height  A,  Varignon  can  retrieve  Torricelli's  law,  h  =  kv2. 
Lagrange l  criticised  this  argument  by  observing  that  the  pressure 
cannot  suddenly  produce  a  finite  velocity.  This  was  a  difficulty  that 
could  not  detain  Varignon,  to  whom  all  force  was  generated  by  velocity. 
But  one  can,  like  Lagrange,  assume  that  the  weight  of  the  column  acts 
on  the  particle  throughout  the  time  that  it  is  leaving  the  vessel.  If  it 
is  then  assumed  that  the  fluid  remains  sensibly  stagnant  in  the  very 
interior  of  the  vessel,  Torricelli's  law  can  be  verified. 

1  MGcanique  analytique,  Section  VI,  part  I  —  Sur  les  principes  de  Vhydrostatique. 


PART  THREE 

THE  ORGANISATION  AND  DEVELOPMENT 

OF  THE  PRINCIPLES  OF  CLASSICAL  MECHANICS 

IN  THE  XVIIIth  CENTURY 


CHAPTER  ONE 

JEAN  BERNOULLI 
AND  THE  PRINCIPLE  OF  VIRTUAL  WORK 

(1717) 

DANIEL  BERNOULLI 
AND  THE  COMPOSITION  OF  FORCES 

(1726) 

1.  JEAN  BERNOULLI  AND  THE  PRINCIPLE  OF  VIRTUAL  WORK. 

Classical  mechanics  was  born  in  the  XVIIth  Century.  The  organ 
isation  and  development  of  the  general  principles  had  still  to  be  ac 
complished — this  was  to  be  the  work  of  the  XVIIIth  Century. 

The  achievements  of  Galileo,  Huyghens  and  Newton  appear  rather 
as  disjointed  parts  than  as  the  continuous  development  of  a  single 
discipline.  Their  successors,  on  the  other  hand,  were  to  participate 
in  a  collective  labour  which,  in  the  hands  of  Lagrange,  was  to  end 
in  an  ordered  science  whose  form  approached  perfection. 

In  the  preceding  parts  of  this  book  we  have  treated  each  author 
in  isolation  from  his  contemporaries,  and  have  attempted  to  follow 
the  chronological  order.  In  order  to  analyse  the  collective  work 
of  the  XVIIIth  Century,  it  will  be  more  satisfactory  if  we  devote 
each  chapter  to  an  attempt  to  collect  together  the  work  of  different 
men  that  was  relevant  to  one  single  topic. 

Although,  at  the  end  of  the  XVIIth  Century,  Varignon  had  tried 
to  found  statics  on  the  one  law  of  the  composition  of  forces,  we  see  Jean 
Bernoulli,  in  a  letter  to  Varignon  himself  (January  26th,  1717),  taking 
up  the  generalisation  of  what  was  really  the  principle  of  virtual  work. 
We  have  seen  that  this  principle  had  been  used  implicitly  as  early 
as  the  Xlllth  Century,  by  the  School  of  Jordanus,  and  that  later  it 
had  been  affirmed  by  Descartes  and  Wallis. 

Jean  Bernoulli  wrote,  in  the  letter  to  Varignon,  "  Imagine  several 
different  forces  which  act  according  to  different  tendencies  or  in  different 


232 


THE   PRINCIPLES  OF  CLASSICAL  MECHANICS 


directions  in  order  to  hold  a  point,  a  line,  a  surface  or  a  body  in  equi 
librium.  Also,  imagine  that  a  small  motion  is  impressed  on  the  whole 
system  of  these  forces.  Let  this  motion  be  parallel  to  itself  in  any 
direction,  or  let  it  be  about  any  fixed  point.  It  will  be  easy  for  you 
to  understand  that,  by  this  motion,  each  of  the  forces  will  advance 
or  recoil  in  its  direction  ;  at  least  if  one  or  several  of  the  forces  do  not 
have  their  tendency  perpendicular  to  that  of  the  small  motion,  in 
which  case  that  force  or  those  forces  will  neither  advance  nor  recoil. 
For  these  advances  or  recoils,  which  are  what  I  call  virtual  velocities, 
are  nothing  else  than  that  by  which  each  line  of  tendency  increases  or 
decreases  because  of  the  small  motion.  And  these  increases  or  decreases 
are  found  if  a  perpendicular  is  drawn  to  the  extremity  of  the  line  of 
tendency  of  any  force.  This  perpendicular  will  cut  ofi"  a  small  part 
from  the  same  line  of  tendency,  in  the  neighbourhood  of  the  small 
motion,  which  will  be  a  measure  of  the  virtual  velocity  of  that  force. 


f 


C 


Fig.  83 


44  For  example,  let  P  be  any  point  in  a  system  which  maintains 
itself  in  equilibrium.  Let  F  be  one  of  the  forces,  which  would  push 
or  draw  the  point  P  in  the  direction  FP  or  PF.  Let  Pp  be  a  short 
straight  line  which  the  point  P  describes  in  a  small  motion,  by  which 
the  tendency  FP  assumes  the  position  fp.  Either  this  will  be  exactly 
parallel  to  FP,  if  the  small  motion  is,  at  every  point,  parallel  to  a  straight 
line  whose  position  is  given ;  or  it  will  make  an  infinitely  small  angle 
with  FP  when  this  is  produced,  and  if  the  small  motion  of  the  system 
is  made  around  a  fixed  point.  Therefore  draw  PC  perpendicular 
to  fp  and  you  will  have  Cp  for  the  virtual  velocity  of  the  force  F,  so 
that  F  X  Cp  is  what  I  call  the  energy. 

44  Observe  that  Cp  is  either  positive  or  negative.  The  point  P  is 
pushed  by  the  force  F.  It  is  positive  if  the  angle  FPp  is  obtuse  and 


JEAN   AND   DANIEL  BERNOULLI  233 

negative  if  the  angle  FPp  is  acute.  But  on  the  contrary,  if  the  point 
P  is  pulled,  Cp  will  be  negative  when  the  angle  FPp  is  obtuse  and 
positive  when  it  is  acute.  All  this  being  understood,  I  form  the  follow 
ing  general  proposition. 

"  In  all  equilibrium  of  any  forces,  in  whatever  way  they  may  be 
applied  and  in  whatever  direction  they  may  act — through  intermedia 
ries  or  directly — the  sum  of  the  positive  energies  will  be  equal  to  the 
sum  of  the  negative  energies  taken  positively.  " 

Jean  Bernoulli's  statement  is  much  more  general  than  those  of 
his  predecessors.  Nevertheless,  it  must  be  remarked  that  the  virtual 
displacements  that  are  contemplated  reduce  to  translations  or  rotations, 
to  displacements  in  which  the  system  behaves  as  a  solid.  Displacements 
of  this  kind  are  not  necessarily  compatible  with  the  constraints  of 
the  system — they  do  not  necessarily  include  the  most  general  virtual 
displacement  which  is  compatible  with  the  constraints. 

Jean  Bernoulli's  principle  does  not  seem  to  have  accomplished 
a  modification  of  Varignon's  point  of  view.  The  latter  was  content 
to  verify  the  principle  in  a  large  number  of  examples,  which  he  treated 
with  the  methods  to  which  he  was  accustomed. 


2.  DANIEL  BERNOULLI  AND  THE  COMPOSITION  OF  FORCES. 

In  a  memoir  which  appeared  in  1726,  called  Examen  principiorum 
mechanicae  et  demonstrationes  geometricae  de  compositione  et  resolutione 
virium,  Daniel  Bernoulli  set  out  to  show  that  the  law  of  the  composition 
of  forces  was  of  necessary,  and  not  of  contingent,  truth.  We  shall 
find  that  Euler  and  d'Alembert  had  similar  preoccupations  in  other 
fields.  The  search  for  such  a  separation  of  purely  rational  truths 
from  those  which  are  subject  to  the  uncertainties  of,  and  correction 
by,  experiment,  was  ever  present  in  learned  minds  throughout  the 
XVIIIth  Century. 

The  question,  by  its  nature,  is  illusory.  But  the  influence  of 
Bernoulli's  demonstration  remained  alive,  and  even  Poisson  was  sub 
ject  to  it  in  1833. 

Bernoulli  regarded  the  hypothesis  of  the  composition  of  motions 
on  which  Varignon  had  based  his  statics  to  be  of  a  contingent  kind. 
But  a  necessary  truth  can  arise  from  two  contingent  hypotheses.  In 
particular,  the  necessary  law  of  the  composition  of  forces  depends, 
not  only  on  the  contingent  hypothesis  of  the  proportionality  of  the 
forces  to  the  velocities  that  they  produce  but  also,  on  the  following 
hypothesis —  A  force  which  acts  on  a  body  that  is  already  moved  by 


234  THE   PRINCIPLES  OF  CLASSICAL  MECHANICS 

another  force  impresses  the  same  velocity  on  the  body  as  if  the  latter 
were  at  rest. 

Basically,  the  development  of  Bernoulli's  demonstration  is  the 
following  x — 

Hypothesis  I.  —  The  composition  of  forces  is  associative. 

Hypothesis  II.  —  The  composition  of  two  forces  in  the  same  direction 
reduces  to  algebraic  addition. 

Hypothesis  III.  —  The  resultant  of  two  equal  forces  is  directed 
along  their  internal  bisector —  "  a  metaphysical  axiom  that  must 
be  regarded  as  a  necessary  truth.  " 

With  this  basis,  Bernoulli  shows  that  if  three  forces  are  in  equilibrium, 
so  too  are  three  forces  which  are  the  multiples  of  the  first  by  the  same 
number.  He  then  establishes  that  the  resultant  of  two  equal  forces 
at  right  angles  is  the  diagonal  of  the  square  that  has  these  two  forces 
as  sides. 

He  continues  with  a  consideration  of  two  unequal  rectangular 
forces  and  finds  that  the  resultant  is  equal  to  the  diagonal  of  the  rect 
angle  of  which  these  two  components  form  two  sides.  He  also  discusses 
the  direction  of  the  resultant. 

Bernoulli   then   treats   pairs    of  components   forming   a   rhombus 


whose  angle  is  equal  to  I  —  I  I  —  L     Then,  in  order,  components  forming 

\2  /  \2/ 
any  rhombus,  a  rectangle,  and  a  parallelogram. 

1  For  further  details,  c/.  JOUGUET,  L.  M.,  Vol.  II,  p.  58. 


CHAPTER  TWO 


THE  CONTROVERSY  ABOUT  LIVING  FORCES 


We  know  that*  as  early  as  1686,1  in  criticising  the  Cartesian  notion 
of  the  conservation  of  quantities  of  motion,  Leibniz  had  suggested 
that  the  "  force  "  acquired  by  a  body  falling  freely  should  be  evaluated 
by  the  height  to  which  this  body  could  rise.  Thus  a  body  whose 
velocity  is  twice  that  of  another  is  endowed  with  a  force  that  is  four 
times  a  great. 

The  Abbe  de  Catelan  protested  that  the  body  effected  this  ascent 
in  twice  the  time.  To  produce  a  quadruple  effect  in  twice  the  time 
is  not  to  have  a  quadruple  force,  but  only  one  which  is  twice  as  great. 
A  child,  in  time,  and  bit  by  bit,  will  carry  a  sack  of  corn  weighing  240 
pounds.  All  force  will  be  infinite  if  no  regard  is  paid  to  time. 

After  much  hesitation,  Jean  Bernoulli  came  round  to  the  opinion 
of  Leibniz.  In  1724  the  Academie  des  Sciences,  without  using  the  words 
living  force,  chose  the  subject  of  the  communication  of  motion  for 
competition. 

Father  Maziere,  an  adversary  of  the  doctrine  of  living  forces,  was  the 
successful  competitor,  in  spite  of  a  contribution  from  Jean  Bernoulli  that 
defended  Leibniz.  In  this  debate  MacLaurin,  Stirling  and  Clarke  were 
opposed  by  the  supporters  of  Leibniz — s'Gravesande,  Wolf  and  Bulfinger. 

Bernoulli  believed  that  the  law  v  =  Js\/Ti  was  related  to  that  of 
gravity  and  that  it  was  not  an  independent  a  priori  law.  Bodies 
would  rise  to  infinity  if  no  cause  prevented  them.  The  limitation 
is  due  to  gravity,  whose  reiterated  obstacles  consumed  a  body's  force 
of  ascent.  Bernoulli  made  use  of  other  examples,  of  which  the  following 
is  typical. 

If  a  perfectly  elastic  sphere  A,  moving  with  the  velocity  AC,  collides 
obliquely  with  an  identical  sphere  which  it  projects  in  the  direction  CD, 
the  body  C  will  be  displaced  on  CD  with  the  velocity  CD  =  BC,  while 
the  body  A  will  continue  its  journey  with  the  velocity  CE  =  CB. 

1  See  above,  p.  219. 


236 


THE   PRINCIPLES  OF  CLASSICAL   MECHANICS 


Now  the  sum  of  the  forces  after  the  impact  must  be  the  same  as  the  sum 
of  the  forces  before  the  impact.  This  would  be  impossible  if  the  force 
were  proportional  to  the  velocity,  for  CE  +  CD  >•  AC.  On  the  other 
hand,  this  relation  is  verified  if  the  force  is  proportional  to  the  square 
of  the  velocity,  for  AC*  =  CD2  + 


A 


B 


C 


Fig.  84 


In  a  Dissertation  of  the  Estimation  and  the  Measurement  of  the  Motive 
Forces  of  Bodies  (1728)  de  Mairan,  like  the  Abbe  de  Catelan,  opposed 
the  evaluation  of  the  force  that  the  followers  of  Leibniz  had  suggested. 
His  premises  were  simple. 

"  As  soon  as  I  conceive  that  a  body  may  be  in  motion,  I  conceive 
of  a  force  that  makes  it  move  [to  be  understood  as  the  vis  motrix 
or  the  force  of  a  body  in  motion,  and  not  the  corresponding  dead  force, 
which  is  zero  for  uniform  motion].  A  uniform  motion  can  never 
indicate  to  us  another  measure  of  the  force  than  the  product  of  the 
simple  velocity  and  the  mass,  " 

Here  is  the  argument — 

"  A  massive  body  having  two  units  of  velocity  is  in  such  a  state 
that  it  can  mount  to  a  height  that  is  four  times  as  great  as  that  to 
which  a  body  with  only  one  unit  of  velocity  would  mount. 

"  This  proportion  implies  common  measure.  This  common  measure 
is  the  time  ;  at  least  I  can  take  the  time  or  the  times  to  be  equal.  .  .  . 

"  Now  given  this,  in  the  effects  of  a  body  which  has  twice  as  much 
velocity,  I  only  find  an  effect  which  is  double  and  not  quadruple — 
a  distance  travelled  which  is  double,  and  a  displacement  of  matter 
which  is  double,  in  equal  times.  From  which  I  conclude,  by  the  very 
principle  of  the  proportionality  of  causes  to  their  effects,  that  the  Motive 


THE    CONTROVERSY   ABOUT   LIVING   FORCES  237 

Force  is  not  quadruple  but  only  double,  as  the  simple  velocity  and  not 
the  square  of  the  velocity.  " 

And  de  Mairan  adds,  "  Strictly  speaking,  the  concept  of  motion 
only  includes  uniformity.  All  motion  should,  on  its  own,  be  uniform, 
just  as  it  should  be  effected  in  a  straight  line ;  the  acceleration  and 
retardation  are  limitations  which  are  foreign  to  its  nature,  as  the  curve 
that  it  is  made  to  describe  is  to  its  proper  direction.  .  .  . 

"  It  is  not  the  distances  travelled  by  the  body  in  retarded  motion 
that  give  the  evaluation  and  the  measure  of  the  motive  force,  but 
rather,  the  distances  which  are  not  travelled,  and  which  should  be  travel 
led,  in  each  instant  by  uniform  motion.  These  distances  which  are 
not  travelled  are  proportional  to  the  simple  velocities.  And  therefore 
the  distances  which  correspond  to  a  retarded  or  decreasing  motive 
force,  in  so  much  as  it  is  consumed  in  its  action,  are  always  proportional 
to  this  force  and  to  the  motion  of  the  body,  just  as  much  in  retarded 
motions  as  in  uniform  motions.  " 

To  explain  this  "  kind  of  paradox,  "  de  Maixan  considers  the 
example  of  two  bodies,  A  and  B,  which  ascend  along  AD  and  Bd.  The 
body  A  has  two  units  of  velocity  and  B  has  only  one. 

"  If  nothing  opposes  its  motive  force,  in  the  first  ,.     ^ 

time  B  will  travel  the  two  toises  Bd  without  losing  T 

any  part  of  this  force  or  any  part  of  the  unit  of  velocity 
which  gives  rise  to  it.  But  because  the  contrary 
impulsions  of  the  heaviness,  which  are  continually 
applied  to  it  succeed  in  consuming  this  force  and 
its  velocity,  and  in  completely  stopping  it,  the  body 
will  only  travel  one  toise  in  its  retarded  motion. 

"  In  the  same  way,  A  would  travel  four  toises 
in  the  first  instant.     The  impulsions  of  the  heaviness  g' 

make  it  fall  back  through  one  toise,  so  that  it  only 
travels  three.  These  impulsions  have  consumed  one  unit  of  force 
and  one  unit  of  velocity,  as  for  B.  But  A  remains  with  one  unit  and,  at 
C,  it  finds  itself  in  the  initial  case  of  B.  It  therefore  has  what  it  needs  to 
travel  the  two  toises  CE.  But  the  impulsions  of  the  heaviness  oppose  it 
and  it  only  travels  CD,  being  pulled  back  through  the  one  toise  ED." 

Thus  the  distance  which  is  not  travelled  by  B  in  the  first  instant 
is  fid.  In  the  first  instant  the  distance  not  travelled  by  A  is  CD,  and 
in  the  second,  is  DJ?. 

This  discussion  is  interesting — its  metaphysical  content  is  so  apparent 
that  we  shall  not  emphasise  it. 

Supporters  and  adversaries  of  the  doctrine  of  living  forces  opposed 
each  other  with  examples  of  impacting  bodies. 


0 


238  THE   PRINCIPLES  OF  CLASSICAL  MECHANICS 

Thus  Herman  considers  a  perfectly  elastic  body  M,  of  mass  1  and 
velocity  2,  colliding  with  a  motionless  sphere  N  of  mass  3.  The 
body  N  will  take,  after  the  impact,  the  velocity  1  while  the  body  M 
will  be  thrown  back  with  the  velocity  1.  If  M  then  meets  a  motion 
less  body  0  of  mass  1,  it  can  communicate  its  velocity  to  the  latter 
and  remain  at  rest.  Therefore  the  force  of  M,  which  has  mass  1  and 
velocity  2,  is  equivalent  to  four  times  the  force  of  a  body  of  mass  1 
and  velocity  1,  which  verifies  the  law  of  living  forces  and  contradicts 
that  of  quantities  of  motion. 

De  Mairan  observed  that  this  coincidence  was  accidental  and 
stemmed  from  the  equality  2  +  2  =  2x2.  For  his  part,  he  considered 
a  body  M  of  mass  1  and  velocity  4  which  he  arranged  to  collide  with  a 
body  N  of  mass  3  which  was  initially  at  rest.  If  M  communicates 
a  velocity  2  to  JV,  the  force  of  N  is  as  6.  The  body  M,  which  keeps 
the  velocity  2,  can  transfer  this  to  a  body  0  of  mass  1,  initially  at  rest. 
The  total  force  of  M  is  therefore  as  6  +  2  =  8,  and  not  as  16  as  the 
law  of  living  forces  would  require. 

The  Marchioness  of  Chatelet  came  round  to  the  doctrine  of  living 
forces  rather  late  in  the  day,  and  added  an  erratum  to  her  book  on  the 
nature  of  fire  (1740).  While  Koenig  was  a  supporter  of  Leibniz, 
Maupertuis  and  Clairaut  remained  indifferent  to  this  controversy. 
In  the  meantime,  de  Mairan  tried  to  convince  the  Marchioness  of  Cha 
telet  and,  in  1741,  Voltaire  himself  proclaimed  his  doubts  about  the 
measure  of  living  forces. 

The  error  of  the  Cartesians,  which  was  corrected  in  the  course 
of  the  controversy  by  de  Mairan,  was  that  of  reckoning  the  quantity 
of  motion  as  m|t;|,  without  regard  to  the  direction  of  the  velocities. 
The  reader  will  easily  verify,  in  all  the  examples  which  have  been 
cited — which  are  examples  of  elastic  impact — that  if  the  direction 
is  introduced,  that  is,  if  quantities  of  motion  mv  are  considered,  then 
the  quantities  £mt;  and  £mt;2  a*6  both  conserved.  Therefore  the  con 
troversy  of  living  forces  was  based  on  a  mis-statement  of  the  doctrine. 
It  rested  on  a  misunderstanding  concerning  the  definition  of  quantity 
of  motion  which,  as  d'Alembert  observed,  divided  the  geometers  for 
more  than  thirty  years. 


CHAPTER  THREE 


EULER  AND  THE  MECHANICS  OF  A  PARTICLE 

(1736) 


Euler  (1707-1783)  was  concerned  with  all  branches  of  dynamics, 
and  we  shall  have  occasion  to  return  to  his  work  in  different  connec 
tions.  For  the  moment,  we  shall  confine  ourselves  to  the  basic  ideas 
of  his  treatment  of  the  dynamics  of  a  particle.  This  is  found  in 
Mechanic^  sive  motus  scientia  analytics  exposita  which  was  published 
in  1736. 

The  very  title  is  a  programme.  Euler  had  read  the  great  creators 
of  mechanics,  especially  Huyghens  and  Newton,  and  he  set  out  to 
fashion  mechanics  into  a  rational  science  by  starting  from  definitions 
and  logically  ordered  propositions.  He  tried  to  demonstrate  the  laws 
of  mechanics  in  such  a  way  that  it  would  be  clear  that  they  were  not 
only  correct,  but  also  necessary  truths. 

To  Euler,  power  or  force  is  characterised  by  the  modification  of 
the  motion  of  a  particle  that  is  produced  by  it.  A  power  acts  along 
a  definite  direction  at  each  instant.  This  is  what  Euler  expresses 
in  the  following  definitions. 

"  Potentia  est  vis  corpus  vel  ex  quiete  in  motum  perducens,  vel  motum 
ejus  alter ans.  " 

"  Directio  potentiae  est  linea  recta  secundum  quam  ea  corpus  movere 
conatur.  " 

In  passing  we  remark  that,  in  Eider's  work,  the  term  "  corpus  " 
denotes  a  particle. 

In  the  absence  of  force  a  particle  either  remains  at  rest,  or  is  animated 
with  a  rectilinear  and  uniform  motion.  Euler  expresses  this  principle 
with  the  help  of  the  concept  of  "  force  of  inertia.  " 

"  Vis  inertiae  est  ilia  in  omnibus  corporibus  insita  facultas  vel  in 
quiete  permanendi  vel  motum  uniformiter  in  directum  continuendi.  " 

Euler  believes  that  "  the  comparison  and  the  measurement  of 
different  powers  should  be  the  task  of  Statics.  "  Euler's  dynamics 


240  THE    PRINCIPLES  OF  CLASSICAL   MECHANICS 

is  therefore  primarily  based  on  the  notion  of  force,  which  he  borrows 
directly  from,  statics  in  accordance  with  Galileo's  procedure. 

Euler  attempted  to  show  that  the  composition  or  the  equivalence 
of  forces  in  statics  could  be  extended  to  their  dynamical  effects.  In 
fact,  he  was  here  concerned  with  a  postulate.  He  also  distinguished 
between  absolute  powers,  such  as  gravity,  that  acted  indifferently 
on  a  body  at  rest  or  in  motion,  and  relative  powers,  whose  effects 
depended  on  the  velocity  of  the  body.  As  an  example  of  such 
a  power,  he  cited  the  force  exerted  by  a  river  on  a  body — this 
force  disappears  when  the  velocity  of  the  body  is  the  same  as  that 
of  the  river. 

In  order  to  determine  the  effect  of  a  relative  power,  an  absolute 
power  is  associated  with  it,  at  least  when  the  body  has  a  known 
velocity. 

We  return  to  the  vis  inertiae  in  the  sense  that  Euler  used  it.  For 
any  body,  this  is  proportional  to  the  quantity  of  matter  that  the  body 
contains. 

64  The  force  of  inertia  is  the  force  that  exists  in  every  body  by  means 
of  which  that  body  persists  in  its  state  of  rest  or  of  uniform  motion 
in  a  straight  line.  It  should  therefore  be  reckoned  by  the  force  or 
power  that  is  necessary  to  take  the  body  out  of  its  state.  Now  different 
bodies  are  taken  out  of  their  state  to  similar  extents  by  powers  which 
are  proportional  to  the  quantities  of  matter  that  they  contain.  There 
fore  their  forces  of  inertia  are  proportional  to  these  powers,  and  conse 
quently,  to  the  quantities  of  matter.  "  Euler  assigns  the  same  vis 
inertiae  to  one  body,  whether  it  is  at  rest  or  in  motion.  For  in  both 
cases  the  body  is  subject  to  the  same  action  and  the  same  absolute 
power. 

Here  we  see  a  systematisation  of  Newtonian  ideas.  Basically 
Euler  introduces  the  mass — in  the  guise  of  a  logical  deduction — by 
means  of  the  physical  assertion  of  proportionality  between  the  powers 
necessary  to  produce  a  given  effect  and  the  quantities  of  matter. 

As  an  example  of  Euler's  analysis,  we  shall  give  his  treatment  of 
the  following  problem. 

"  Proposition  XIV.  —  Problem.  —  Being  given  the  effect  of  an 
absolute  power  on  a  particle  at  rest,  to  find  the  effect  of  the  same  power 
on  the  same  particle^  when  the  latter  is  moving  in  any  way.  " 

The  absolute  power  which  is  given  will  make  a  body  A,  initially 
at  rest,  travel  the  path  dz  =  AC  in  the  time  dt. 

If  A  has  the  velocity  c,  in  the  absence  of  any  power  it  will  travel 
the  path  AB  —  cdt  in  the  time  dt. 

But  the  given  power,  being  absolute,  acts  on  A  in  motion  in  the 


THE   MECHANICS    OF   A  PARTICLE 


241 


same  way  as  it  acts  on  A  at  rest.  Therefore  the  effect  of  the  power 
is  compounded  with  that  of  the  velocity,  and  the  body  A  comes  to 
D,  where  BD  =  AC. 

A 


Fig.  86 

Under  the  effect  of  the  given  power,  the  velocity  of  A  will  become 
AD 


A  simple  geometrical  argument  shows  that 

dc  =  —  cos  BAG. 
dt 

Strictly  speaking  it  would  be  more  natural  to  regard  the  effect 
of  the  power  as  being  the  increase  of  the  velocity  between  the  time 
t  and  the  time  t  +  dt ;  that  is,  to  consider  the  quantity  2dc  instead 
of  the  difference  between  the  initial  velocity  c  and  the  mean  velocity 
of  A  during  the  time  dt.1 

Euler  then  studies  the  effect  of  a  power  B  on  a  body  when  the 
effect  of  a  power  A  on  the  same  body  is  given.  He  concludes — 

"  If  a  body  is  affected  by  many  powers,  at  first  it  may  be  thought 
of  as  divided  into  as  many  parts,  on  each  of  which  one  of  the  powers 
acts.  Then,  when  the  different  parts  have  been  drawn  by  their  respec 
tive  powers  for  an  element  of  time,  it  is  imagined  that  they  suddenly 
unite.  When  this  is  accomplished,  the  position  of  their  reunion  will 
be  that  at  which  the  whole  body  would  have  arrived  in  the  same  time 
by  the  simultaneous  action  of  all  the  powers.  The  truth  of  this  princi 
ple  can  be  illustrated  by  remarking  that  the  parts  of  a  body  can  be 
held  together  by  very  strong  springs  which  though  they  act  in  an 
undefined  manner,  can  be  supposed  to  relax  completely  in  the  interval 


1  C/.  JOUGUET,  L.  M.,  Vol.  II,  p.  43. 


16 


242  THE   PRINCIPLES  OF  CLASSICAL  MECHANICS 

of  time,  and  to  contract  suddenly  with  an  infinite  force,  afterwards, 
in  such  a  way  that  the  conjunction  of  the  separated  parts  takes  no 
time.  " 

Thus  Euler's  law  of  dynamics  takes  the  form — 

The  increase,  dc,  of  the  velocity  is  proportional  to  pdt,  where  p  is  the 
power  acting  on  the  body  during  the  time  dt.  This  applies  to  a  single 
body ;  if  several  bodies  are  considered  simultaneously,  their  masses 
must  be  introduced. 

Therefore  this  law  emphasises  the  impulse  of  the  force  during  an 
elementary  time,  or  the  impulse  that  gives  rise  to  an  increase  of 
momentum. 

Euler  declared  that  this  law  was  not  only  true,  but  also  a  necessary 
truth,  and  that  a  law  identifying  mdc  as  p*dt  or  as  p^dt  would  imply 
a  contradiction.  Clearly  this  is  an  illusion  of  the  author. 

Eider's  treatise  then  continues  with  a  study  of  a  large  number 
of  problems.  First  he  treats  a  free  particle,  and  concludes  with  a 
particle  bound  on  a  curve  or  a  surface,  either  in  a  vacuum  or  a  resisting 
medium.  His  work  was  the  first  to  merit,  for  the  order  and  the  pre 
cision  of  its  demonstrations,  the  name  of  a  treatise  of  rational  mechanics. 


CHAPTER  FOUR 


JACQUES  BERNOULLI 
AND  THE  CENTRE  OF  OSCILLATION 

(1703) 

D'ALEMBERT'S  TREATISE  ON  DYNAMICS 

(1743) 


1.  JACQUES  BERNOULLI  AND  THE  CENTRE  OF  OSCILLATION. 

In  1703  Jacques  Bernoulli  returned  to  the  famous  problem  of 
the  search  for  a  centre  of  oscillation,  and  gave  a  solution  of  it  which 
contained  the  germ  of  d'Alembert's  principle.  Jacques  Bernoulli's 
paper  was  called  "  (General  demonstration  of  the  centre  of  balancing 
and  of  oscillation  deduced  from  the  nature  of  the  lever.  " 

He  considers  a  lever  which  is  free  to  turn  about  a  point  A  and  whose 
different  arms  carry  weights  or  powers  which  act  perpendicularly  to 
the  arms.  If  the  powers  are  divided  into  two  groups  that  act  on  the 
lever  in  opposite  senses,  and  if  the  sum  of  the  products  of  the  arms 
of  the  lever  and  the  powers  has  the  same  absolute  value  for  each  group, 
then  the  lever  will  remain  in  equilibrium.  This  had  been  shown  by 
Mariotte  in  the  Treatise  on  the  percussion  of  bodies. 

Given  this,  let  A  represent  the  axis  of  suspension,  and  let  AC  and 
AD  join  A  to  two  arbitrary  elements  of  a  compound  pendulum  (for 
simplicity  assumed  to  be  plane).  Then  let  AM  be  the  simple  pendulum 
isochronous  with  the  compound  pendulum.  ^ 

Consider  the  motion  of  the  elements  C,  D  and  M  of  the  compound 
pendulum.  Their  velocities  are  proportional  to  AC^  AD  and  AM. 
At  each  instant  the  gravity  adds  an  impact  or  an  impulse  which  is 
represented  by  MN9  CO,  DP,  "  short  vertical  and  equal  lines.  "  Take 
NK,  OT  and  PV  perpendicular  to  the  arcs  MK,  CT,  DV. 

Bernoulli  considers  the  "  motions  "  MZV,  CO,  DP  as  being  decom 
posed  into  motions  MK  and  KN ;  CT  and  TO  ;  DV  and  VP.  The 
motions  JfiCJV,  TO,  VP  "  distribute  themselves  over  the  whole  axis  A 


244 


THE   PRINCIPLES   OF  CLASSICAL   MECHANICS 


and  there  lose  themselves  completely.  "  Because  of  the  isochronism 
of  the  points  C,  D  and  M,  the  motions  MK,  CT  and  VD  suffer  "  some 
change.  "  If,  for  example,  M  comes  to  K  (without  alteration),  then  C 
comes  to  R  and  D  to  S,  and  the  arcs  MK,  CR  and  DS  will  be  similar. 
The  effort  of  gravity  acting  on  the  point  C  is  not  exhausted  at  JR  and 
"  the  remainder,  RT,  must  be  used  to  push  the  body  D  along  VS.  " 
But  D  itself  resists  as  much  as  it  is  pushed,  and  everything  happens  as 
if  D  travelled  to  S — as  if  there  were  a  force  "  which  tries  to  repel  it 
from  S  to  V.  " 


Fig.  87 

To  sum  up,  the  lever  CAD  is  in  equilibrium  under  the  action  of 
weights  like  C,  "  pulling  or  pushing  from  one  side  with  forces  or  velo 
cities  .RT,"  and  weights  like  D,  pulling  or  pushing  in  the  opposite  sense. 

Therefore  Bernoulli  writes  S  (C  X  CA  X  RT)  =  S  (D  X  AD  X 
VS)  and,  from  this,  deduces  the  solution  of  the  problem  of  the  centre 
of  oscillation. 


2.    THE    INTRODUCTORY    ARGUMENT     OF    D'ALEMBERT'S     TREATISE     ON 
DYNAMICS. 

The  first  edition  of  d'Alembert's  Traite  de  dynamique  is  dated  1743. 
Here  we  shall  follow  an  edition  of  1758,  which  was  corrected  and  added 
to  by  the  author. 

In  an  introductory  discussion,  d'Alembert  explains  his  philosophy 


JACQUES   BERNOULLI  AND    D'ALEMBERT  245 

of  mechanics.  The  Sciences  are  divided  into  two  groups — those  which 
are  based  on  principles  which  are  necessarily  true  and  clear  in  themselves ; 
and  those  which  are  based  on  physical  principles,  experimental  truths, 
or  simply  on  hypotheses.  Mechanics  belongs  to  the  first  category  of 
purely  rational  sciences,  although  it  appears  to  us  as  less  direct  than 
Geometry  and  Algebra.  It  has  failed  to  clarify  the  mystery  of  impene 
trability,  the  enigma  of  the  nature  of  motion,  and  the  metaphysical 
principle  of  the  laws  of  impact.  .  .  . 

The  best  method  of  discussing  any  part  of  mathematics  "  is  to 
regard  the  particular  subject  of  that  science  in  the  most  abstract  and 
direct  way  possible  ;  to  suppose  nothing,  and  to  assume  nothing  about 
that  subject,  that  the  properties  of  the  science  itself  does  not  suppose.  " 

D'Alembert  sets  out  "  to  throw  back  the  boundaries  of  mechanics 
and  to  smooth  out  the  approach  to  it  ...  and,  in  some  way,  to  achieve 
one  of  these  objects  by  means  of  the  other.  That  is,  not  only  to  deduce 
the  principles  of  mechanics  from  the  clearest  concepts,  but  also  to  apply 
them  to  new  ends.  "  He  strives  "  to  make  everything  clear  at  once  ; 
both  the  futility  of  most  of  the  principles  that  have  so  far  been  used  in 
mechanics,  and  the  advantage  that  can  be  obtained  from  the  combin 
ation  of  others,  for  the  progress  of  that  Science.  In  a  word,  to  extend 
the  principles  and  reduce  them  in  number.  " 

The  nature  of  motion  has  been  much  discussed.  "  Nothing  would 
seem  more  natural  than  to  conceive  of  it  as  the  successive  application, 
of  the  moving  body  to  the  different  parts  of  infinite  space.  "  But  the 
Cartesians,  "  a  faction  that,  in  truth,  now  barely  exists,  "  refuse  to 
distinguish  space  from  matter.  In  order  to  counter  their  objections, 
d'Alembert  makes  a  distinction  between  impenetrable  space,  provided 
by  what  are  properly  called  bodies,  and  space  pure  and  simple,  penetr 
able  or  not,  which  can  be  used  to  measure  distances  and  to  observe 
the  motion  of  bodies. 

"  The  nature  of  time  is  to  run  uniformly,  and  mechanics  supposes 
this  uniformity.  "  This  is  Newtonian. 

"A  body  cannot  impart  motion  to  itself, "  There  must  be  an  extern 
al  cause  in  order  to  move  it  from  rest.  But  "  if  the  existence  of 
motion  is  once  supposed,  without  any  other  particular  hypothesis,  the 
most  simple  law  that  a  moving  body  can  observe  in  its  motion  is  the 
law  of  uniformity,  and  consequently,  this  is  that  which  it  must  conform 
to.  ...  Therefore  motion  is  inherently  uniform.  " 

D'Alembert  defines  the  force  of  inertia  as  the  property  of  bodies  of 
remaining  in  their  state  of  rest  or  motion.  Among  the  means  that  can 
alter  the  motion  of  a  body,  apart  from  constraints,  he  only  allows 
two — impact  (or  impulse)  and  gravity  (or,  more  generally,  attraction). 


246  THE   PRINCIPLES  OF  CLASSICAL   MECHANICS 

In  this  connection  it  seems  that  d'AIembert  criticises  the  very  prin 
ciple  of  Euler's  mechanics. 

"  Why  have  we  gone  back  to  the  principle,  which  the  whole  world 
now  uses,  that  the  accelerating  or  retarding  force  is  proportional  to  the 
element  of  the  velocity  ?  A  principle  supported  on  that  single  vague  and 
obscure  axiom  that  the  effect  is  proportional  to  its  cause. 

"  We  shall  in  no  way  examine  whether  this  principle  is  a  necessary 
truth  or  not.  We  only  say  that  the  evidence  that  has  so  far  been  pro 
duced  on  this  matter  is  irrelevant.  Neither  shall  we  accept  it,  as  some 
geometers  have  done,  as  being  of  purely  contingent  truth,  which  would 
destroy  the  exactness  of  mechanics  and  reduce  it  to  being  no  more  than  an 
experimental  science.  We  shall  be  content  to  remark  that,  true  or  false, 
clear  or  obscure,  it  is  useless  to  mechanics  and  that,  consequently,  it 
should  be  abolished.  " 

This  shows  in  what  sense  d'AIembert  interpreted  the  task  of  making 
mechanics  into  a  rational  science,  and  the  extent  to  which  he  valued  his 
own  principle. 

D'AIembert  made  appeal  to  a  principle  of  the  composition  of  motions, 
of  which  he  intended  to  give  simple  evidence. 

When  a  body  changes  in  direction,  its  motion  is  made  up  of  the 
initial  motion  and  an  acquired  motion.  Conversely,  the  initial  motion 
can  be  compounded  of  a  motion  which  is  assumed  and  a  motion  which 
is  lost. 

D'AIembert  established  the  laws  of  motion  in  the  presence  of  any 
obstacle  in  the  following  way.  The  motion  of  the  body  before  meeting 
the  obstacle  is  decomposed  into  two  motions —  one  which  is  unchanged, 
and  another  which  is  annihilated  by  the  obstacle. 

If  the  obstacle  is  insurmountable,  the  laws  of  equilibrium  are  used. 
These  laws  are  expressed  by  a  relation  of  the  kind 


m  v 

mr  v 


where  v,  v'  are  the  velocities  with  which  the  masses  m,  m!  tend  to  move. 
Only  when  there  is  perfect  symmetry,  or  when 

m  =  m'  v'  =  —  v 

does  the  problem  appear  inherently  clear  and  simple  to  d'AIembert, 
and  he  tries  to  reduce  all  other  situations  to  this  one.  We  have  seen 
that  this  was  an  illusion  which  Archimedes  had  in  his  investigation  of 
the  equilibrium  of  the  lever. 

And  d'AIembert  concludes,  "  The  principle  of  equilibrium,  together 
with  the  principles  of  the  force  of  inertia  and  of  compound  motion, 


JACQUES   BERNOULLI   AND   D'ALEMBERT  247 

therefore  leads  us  to  the  solution  of  all  problems  which  concern  the 
motion  of  a  body  in  so  far  as  it  can  be  stopped  by  an  impenetrable  and 
immovable  obstacle — that  is,  in  general,  by  another  body  to  which  it 
must  necessarily  impart  motion  in  order  to  keep  at  least  a  part  of  its 
own.  From  these  principles  together  can  easily  be  deduced  the  laws 
of  the  motion  of  bodies  that  collide  in  any  manner  whatever,  or  which 
affect  each  other  by  means  of  some  body  placed  in  between  them  and 
to  which  they  are  attached.  " 

Lagrange  said,  and  it  is  often  repeated,  that  d'Alembert  had  reduced 
dynamics  to  statics  by  means  of  his  principle.  The  last  quotation  shows 
clearly  that  d'Alembert  himself  did  not  accept  such  a  simple  interpreta 
tion.  On  the  contrary,  he  stressed  the  fact  that  "  the  three  principles 
of  the  force  of  inertia,  of  compound  motion  and  of  equilibrium  are 
essentially  different  from  each  other.  " 

D'Alembert's  beliefs  are  thus  clearly  expressed  in  the  first  pages  on 
his  introduction.  But  he  also  made  clear  his  view  on  the  problems 
which  were  popular  in  his  time.  Above  all,  he  intended  to  take  account 
of  motion  without  being  concerned  with  motive  causes  ;  he  completely 
banished  the  forces  inherent  to  bodies  in  motion,  "  as  being  obscure  and 
metaphysical,  and  which  are  only  able  to  cover  with  obscurity  a  subject 
that  is  clear  in  itself.  " 

This  is  why  d'Alembert  refused  "  to  start  an  examination  of  the 
celebrated  question  of  living  forces,  which  has  divided  the  geometers 
for  thirty  years.  "  To  him,  this  question  was  only  a  dispute  about 
words,  for  the  two  opposing  sides  were  entirely  in  agreement  of  the  fun 
damental  principles  of  equilibrium  and  of  motion.  Their  solutions  of  the 
same  problem  coincided,  "  if  they  were  sound.  " 

D'Alembert  also  discussed  the  question  of  knowing  whether  the  laws 
of  mechanics  are  of  necessary  or  contingent  truth.  This  question  had 
been  formulated  by  the  Academy  of  Berlin. 

In  order  that  this  question  may  have  a  meaning,  it  is  necessary  to 
dispense  with  "  every  sentient  being  capable  of  acting  on  matter,  every 
will  of  intellectual  origin.  "  It  is  said  that  d'Alembert  rejected  every 
finalist  explanation  involving  the  wisdom  of  the  Creator — we  shall 
return  to  this  in  connection  with  the  principle  of  least  action. 

To  d'Alembert,  the  principles  of  mechanics  are  of  necessary  truth. 

"  We  believe  that  we  have  shown  that  a  body  left  to  itself  must 
remain  forever  in  its  state  of  rest  or  of  uniform  motion  ;  that  if  it  tends 
to  move  along  the  two  sides  of  a  parallelogram  at  once,  the  diagonal 
is  the  direction  that  it  must  take  ;  that  is,  that  it  must  select  from  all 
the  others.  Finally,  we  have  shown  that  all  the  laws  of  the  communic 
ation  of  motion  between  bodies  reduce  to  the  laws  of  equilibrium,  and 


248  THE   PRINCIPLES  OF  CLASSICAL   MECHANICS 

that  the  laws  of  equilibrium  themselves  reduce  to  those  of  the  equili 
brium  of  two  equal  bodies  which  are  animated  in  different  senses  by 
equal  virtual  velocities.  In  the  latter  instance,  the  motions  of  the  two 
bodies  evidently  cancel  each  other  out ;  and  by  a  geometrical  conse 
quence,  there  will  also  be  equilibrium  when  the  masses  are  inversely 
proportional  to  the  velocities.  It  only  remains  to  know  whether  the 
case  of  equilibrium  is  unique — that  is,  whether  one  of  the  bodies  will 
necessarily  force  the  other  to  move  when  the  masses  are  no  longer 
inversely  proportional  to  the  velocities.  Now  it  is  easy  to  believe  that 
as  soon  as  there  is  one  possible  and  necessary  case  of  equilibrium,  it 
will  not  be  possible  for  others  to  exist  without  the  laws  of  impact 
— which  necessarily  reduce  to  those  of  equilibrium — becoming  indeter 
minate.  And  this  cannot  be,  since,  if  one  body  collides  with  another, 
the  result  must  necessarily  be  unique,  the  inevitable  consequence  of 
the  existence  and  the  impenetrability  of  bodies.  " 

3.  D'ALEMBERT  AND  THE  CONCEPT  OF  ACCELERATING  FORCE. 

Of  all  the  causes  that  could  influence  a  body,  d'Alembert  was  of  the 
opinion  that  only  impulse  (that  is,  impact)  was  perfectly  determinate. 
All  other  causes  are  entirely  unknown  to  us  and  can  only  be  distinguished 
by  the  variation  of  motion  which  they  produce.  The  "  accelerating 
force  "  9?  is  introduced  by  the  relation  cpdt  =  du,  a  relation  between  the 
time  t  and  the  velocity  u — the  only  observable  kinematic  quantities. 
This  relation  is  the  definition  of  (p. 

Therefore,  to  d'Alembert,  this  force  was  a  derived  concept,  though 
to  Daniel  Bernoulli  and  Euler  it  constituted  a  primary  concept. 

To  Daniel  Bernoulli,  the  law  cpdt  =  du  was  a  contingent  truth  ;  to 
Euler,  a  necessary  truth. 

D'Alembert  wrote,  "  for  us,  without  wishing  to  discuss  here  whether 
this  principle  is  a  necessary  or  a  contingent  truth,  we  shall  be  content 
to  take  it  as  a  definition,  and  to  understand  by  the  phrase  6  accelerating 
force  ',  merely  the  quantity  to  which  the  increase  in  velocity  is  propor 
tional.  "  i 

4.  D'ALEMBERT'S  PRINCIPLE. 

D'Alembert's  principle  was  made  the  subject  of  a  letter  to  the 
Academic  des  Sciences  as  early  as  1742.  In  this  book,  we  shall  follow 
the  presentation  of  the  principle  which  appears  in  the  1758  edition  of 
the  Traite  de  Dynamique  (2nd  Part,  Chapt.  I,  p.  72). 

1  Trait^  de  Dynamique^  cor.  VI,  p.  25  (1758  edition). 


JACQUES    BERNOULLI   AND   D'ALEMBERT  249 

PRESENTATION  OF  THE  PRINCIPLE 

"  Bodies  only  act  on  each  other  in  three  ways  that  are  known  to 
us —  either  by  immediate  impulse  as  in  ordinary  impact ;  or  by  means  of 
some  body  interposed  between  them  and  to  which  they  are  attached  ; 
or  finally,  by  a  reciprocal  property  of  attraction,  as  they  do  in  the 
Newtonian  system  of  the  Sun  and  the  Planets.  Since  the  effects  of  this 
last  mode  of  action  have  been  sufficiently  investigated,  I  shall  confine 
myself  to  a  treatment  of  bodies  which  collide  in  any  manner  whatever, 
and  of  those  which  are  acted  upon  be  means  of  threads  or  rigid  rods. 
I  shall  dwell  on  this  subject  even  more  readily  because  the  greatest 
geometers  have  only  so  far  (1742)  solved  a  small  number  of  problems 
of  this  kind,  and  because  I  hope,  by  means  of  the  general  method  which 
I  am  going  to  present,  to  equip  all  those  who  are  familar  with  the  calcu 
lations  and  principles  of  mechanics  so  that  they  can  solve  the  most 
difficult  problems  of  this  kind. 

DEFINITION 

"  In  what  follows,  I  shall  call  motion  of  a  body  the  velocity  of  this 
same  body  and  shall  take  account  of  its  direction.  And  by  quantity 
of  motion,  I  shall  understand,  as  is  customary,  the  product  of  the  mass 
and  the  velocity. 

GENERAL  PROBLEM 

"  Let  there  be  given  a  system  of  bodies  arranged  in  any  way  with 
respect  to  each  other  ;  and  suppose  that  a  particular  motion  is  imparted 
to  each  of  these  bodies,  which  it  cannot  follow  because  of  the  action  of  the 
other  bodies —  to  find  the  motion  that  each  body  must  take. 

SOLUTION 

"  Let  A9  JB,  C,  etc be  the  bodies  that  constitute  the  system  and 

suppose  that  the  motions  a,  6,  c,  etc.  .  ,  .  are  impressed  on  them  ;  let 
there  be  forces,  arising  from  their  mutual  action,  which  change  these 
into  the  motions  a,  I>,  c,  etc.  ...  It  is  clear  that  the  motion  a  impressed 
on  the  body  A  can  be  compounded  of  the  motion  a  which  it  acquires 
and  another  motion  a.  In  the  same  way  the  motions  fr,  c,  etc.  .  .  .  can 
be  regarded  as  compounded  of  the  motions  1>  and  /?,  c  and  #,  etc.  .  . . 
From  this  it  follows  that  the  motions  of  the  bodies  -4,  B,  C,  etc.  .  .  . 
would  be  the  same,  among  themselves,  if  instead  of  their  having  been 


250  THE   PRINCIPLES   OF  CLASSICAL   MECHANICS 

given  the  impulses  a,  6,  c,  etc.  .  .  .  they  had  been  simultaneously  given 
the  twin  impulsions  a  and  a,  b  and  /?,  c  and  #,  etc.  .  .  .  Now,  by  sup 
position,  the  bodies  A^  B,  C,  etc.  .  .  .  have  assumed,  by  their  own  action, 
the  motions  a,  b,  c,  etc.  .  .  .  Therefore  the  motions  a,  /?,  «,  etc.  .  .  . 
must  be  such  that  they  do  not  disturb  the  motions  a,  b,  c,  etc. ...  in 
any  way.  That  is  to  say,  that  if  the  bodies  had  only  received  the  mo 
tions  a,  /?,  X,  etc.  .  .  .  these  motions  would  have  been  cancelled  out 
among  themselves,  and  the  system  would  have  remained  at  rest. 
44  From  this  results  the  following  principle  for  finding  the  motion  of 
several  bodies  which  act  upon  each  other.  Decompose  each  of  the 
motions  a,  &,  c,  etc.  .  .  .  which  are  impressed  on  the  bodies  into  two  others, 
a  and  a,  b  and  /?,  c  and  X,  etc.  .  .  .  which  are  such  that  if  the  motions 
a,  I),  c,  etc.  .  .  .  had  been  impressed  on  the  bodies,  they  would  have  been 
retained  unchanged  ;  and  if  the  motions  a,  /?,  X,  etc.  .  .  .  alone  had  been 
impressed  on  the  bodies,  the  system,  would  have  remained  at  rest.  It  is 
clear  that  a,  b,  c,  etc. .  .  .  will  be  the  motions  that  the  bodies  will  take 
because  of  their  mutual  action.  This  is  what  it  was  necessary  to  find.  " 

5.  D'ALEMBERT'S  SOLUTION  OF  THE  PROBLEM  OF  THE  CENTRE  OF  OSCIL 
LATION. 

Although  d'Alembert's  principle  is  perfectly  clear,  its  application  is 
difficult,  and  the  Traite  de  Dynamique  remains  a  difficult  book  to  read. 

As  a  concrete  example  of  its  application,  we 

C  shall  give  d'Alembert's  solution  of  the  celebrat 

ed  problem  of  the  centre  of  oscillation.1 

46  Problem.  —  To  find  the  velocity  of  a  rod  CR 
fixed  at  C,  and  loaded  with  as  many  weights  as 
may  be  desired,  under  the  supposition  that  these 
bodies,  if  the  rod  had  not  prevented  them,  would 
have  described  infinitely  short  lines  AO,  BQ,  jRT, 
perpendicular  to  the  rod,  in  equal  times. 

"All  the  difficulty  reduces  to  finding  the 
line  RS  travelled  by  one  of  the  bodies,  J?,  in  the 

._  . time  that  it  would  have  travelled  RT.     For  then 

_.     00  the  velocities  J3G,  AM,  of  all  the  other  bodies 

rig.  oo  i 

are  known. 

44  Now  regard  the  impressed  velocities,  RT, 

BQ,  AO  as  being  composed  of  the  velocities  RS  and  ST ;  EG  and 
—  GQ  ;  AM  and  — MO.    By  our  principle,  the  lever  CAR  would  have 

1  Trait^  de  Dynamique,  p.  96. 


JACQUES   BERNOULLI   AND    D'ALEMBERT  251 

remained  in  equilibrium  if  the  bodies  jR,  B,  A  had  received  the  motions 
ST,  —  GQ,  —MO  alone. 

"  Therefore 

A-MO.AC  +  B.QG-BC  =  R.ST-CR. 

"  Denoting  AO  by  a,  BQ  by  6,  J?Tby  c,  (L4  by  r,  CB  by  r',  CjR  by  Q 
and  JRS  by  2,  we  will  have 


R  (c-*)  Q=Ar-  —  a    +  Br'    —  - 

\Q  /  \  Q 

"  Consequently 

_  Aarq  +  Bbr'g  +  RCQ* 

*~~      Ar*  +  Br'2  +  RQ*    " 

"  Corollary.  —  Let  .F,/,  <p  be  the  motive  forces  of  the  bodies  A,  B, 
The  accelerating  force  will  be  found  to  be 

Fr+fr'  +  w 

Ke 


+  Br2  +RQ 

-  .  F  f  <P 

on  giving  a,  6,  c,  their  values  -?,  ~,  —  *     Therefore,  if  the  element  of  arc 

joL    O   /x 

described  by  the  radius  CR  is  taken  to  be  ds  and  the  velocity  of  R  to 
be  u,  then,  in  general, 


whatever  the  forces  F,  /,  cp  may  be.     It  is  easy,  by  this  means,  to  solve 
the  problem  of  centres  of  oscillation  under  any  hypothesis. 

6.  THE  PRIORITY  OF  HERMAN  AND  EULER  IN  THE  MATTER  OF  D'ALEM- 
BERT'S  PRINCIPLE. 

After  recalling  Jacques  Bernoulli's  solution  of  the  problem  of  the 
centre  of  oscillation,  d'Alembert  remarks  that  Euler,  in  Volume  III  of  the 
old  Commentaries  of  the  Academy  of  Petersbourg  (1740),  had  used  the 
principle  according  to  which  the  powers  JR-JRS,  B-J3G,  A  -AM  must  be 
equivalent  to  the  powers  JR-JRT,  B-BQ,  A-AO.  "But  M.  Euler  has 
in  no  way  demonstrated  this  principle  and  this,  it  seems  to  me,  can  only 
be  done  by  means  of  ours.  Moreover,  the  author  has  only  applied  this 
principle  to  the  solution  of  a  small  number  of  problems  concerning  the 
oscillation  of  flexible  or  inflexible  bodies,  and  the  solution  that  he  has 
given  to  one  of  these  problems  is  not  correct.  [This  was  the  problem 
of  the  oscillation  of  a  solid  body  on  a  plane.]  This  shows  to  what  extent 


252  THE   PRINCIPLES  OF  CLASSICAL   MECHANICS 

our  principle  is  preferable  for  solving  not  only  problems  of  tbat  kind, 
but  in  general,  all  questions  of  dynamics.  "  1 

Lagrange  had  the  following  comment  to  make  on  this  matter. 

"  If  it  is  desired  to  avoid  the  decompositions  of  motions  that  d'Alem- 
bert's  principle  demands,  it  is  only  necessary  to  establish  immediately 
the  equilibrium  between  the  forces  and  the  motions  they  generate,  but 
taken  in  the  opposite  directions.  For  if  it  is  imagined  that  there  is 
impressed  on  each  body  the  motion  that  it  must  take,  in  the  opposite 
sense,  it  is  clear  that  the  system  will  be  reduced  to  rest.  Consequently, 
it  is  necessary  that  these  motions  should  destroy  those  that  the  body 
had  received  and  which  they  would  have  followed  without  this  inter 
action.  Thus  there  must  be  equilibrium  between  all  these  motions  or 
between  the  forces  which  can  produce  them. 

"  This  method  of  recalling  to  mind  the  laws  of  Dynamics  is  certainly 
less  direct  than  that  which  follows  from  d'Alembert's  principle,  but  it 
offers  greater  simplicity  in  applications.  It  reduces  to  that  of  Herman  2 
and  of  Euler,3  who  used  it  in  the  solution  of  many  problems  in  Mechan 
ics,  and  which  is  found  in  many  treatises  on  mechanics  under  the  name 
of  d? Alember? $  Principle.  " 

However  clear  these  priorities  may  be,  they  do  not  detract  from  the 
originality  of  d'Alembert's  conceptions.  His  work  stands  out  because 
of  its  philosophic  breadth  of  view,  because  of  its  property  of  unifying 
and  generalising,  and  its  equal  is  not  found  among  the  work  of  his 
immediate  predecessors. 

7.  D'ALEMBERT  AND  THE  LAWS  OF  IMPACT. 

D'Alembert  systematically  applied  his  principle  to  the  solution  of 
all  the  problems  which  appear  in  his  Traite,  whether  they  concern  bodies 
which  are  supported  by  threads  or  rods,  bodies  which  oscillate  on  planes, 
bodies  which  interact  by  means  of  threads  on  which  they  can  run  freely, 
or  different  modes  of  impact. 

In  the  problems  of  impact  d'Alembert,  at  first,  only  considers 
*4  hard  bodies  "  (that  is,  bodies  deprived  of  their  elasticity).  Thus,  if 
a  body  of  mass  M  and  velocity  U  collides  directly  with  a  body  of  mass 
m  and  velocity  w,  d'Alembert  writes  the  following  relations  between  the 
velocities. 

u  —  v  +  u  —  v 

u= v+ u-v 

1  TraitS  de  Dynamique^  p.  101. 

2  Phoronomia,  sive  De  viribus  et  motibus  corporum  solidorum  et  fluidorum,  Amster 
dam,  1716. 

3  The  paper  cited  by  d'ALEMBERT  (see  the  beginning  of  this  §). 


JACQUES    BERNOULLI   AND   D'ALEMBERT  253 

Here  v  and  V  are  the  velocities  of  the  first  and  the  second  bodies 
after  the  impact. 

After  the  impact  V  =  v  and  because  of  the  principle,  m(u  —  v)  -j- 

M(U—  V)  =  0.     Therefore  Fand  v  =  *""  +  MU. 

V  '  M  +  771 

D'Alembert  next  deduces  the  laws  of  the  impact  of  elastic  bodies 
from  those  of  the  impact  of  hard  bodies  by  the  following  procedure. 
"  If  as  many  bodies  as  may  be  desired  collide  with  each  other  so  that  when 
it  is  supposed  that  they  are  perfectly  hard  and  without  elasticity,  they  all 
remain  at  rest  after  the  impact  ;  I  say  that  if  they  are  of  perfect  elasticity 
each  one  will  rebound  after  the  impact  with  the  velocity  it  had  before  the 
impact.  For  the  effect  of  the  elasticity  is  to  give  back  to  each  body 
the  velocity  which  it  has  lost  because  of  the  action  of  the  others.  "  l 

Thus  d'Alembert  separated  the  theory  of  impact  from  all  appeal  to 
the  conservation  of  living  forces. 


8.    D'ALEMBERT  AND  THE  PRINCIPLE  OF  LIVING  FORCES. 

D'Alembert  prepared  the  way  for  Lagrange  by  setting  out  to  show 
that  the  principle  of  the  conservation  of  living  forces  was  a  consequence 
of  the  laws  of  dynamics  for  systems  with  restraints  composed  of  threads 
and  inflexible  rods,  just  as  the  laws  of  impact  were  a  consequence  of 
this  same  principle.  Without  giving  a  general  demonstration  of  this 
fact,  d'Alembert  gave  "  the  principles  sufficient  for  obtaining  the  de 
monstration  in  every  particular  case.  " 

We  shall  confine  ourselves  here  to  a  very  simple  case. 

"  Imagine  two  bodies,  A  and  jB,  of  an  infinitely  small  extension,  to 
be  attached  to  an  inflexible  rod  AB.  And  suppose  that  any  directions 
and  velocities  are  imparted  to  these  bodies,  and  that  these  velocities 
are  represented  by  the  infinitely  short  lines  AK  and  J5D.  According  to 
our  principle,  it  is  necessary  to  construct  the  parallelograms  MC  and 
NL9  in  which  LC  =  AB  and  B  X  BM  =  A  X  AN.  The  velocities  and 
the  directions  of  the  bodies  B  and  A  will  be  BC  and  AL.  Now  J3C2  = 
ED*  —  2CE-  CD—  CD*  and  AD  =  Z8?  +  2PL  -  KL  —  KL*.  Therefore 

B.B&  +  A.JI*  =  A.  AK*  +  B  .  HEM-  AQPL-JSL  —  KL*±  — 

B(2CE-  CD  +  CD2),  which  reduces  to  A  •  AK*  +  B  •  BD*—  A  -  KL*  — 
B*CD\  since  CE  =  PL  and  A-KL  =  B-CD. 
"Therefore 


1  Traite  de  Dynamique,  p.  218. 

2  Ibid.,  p  253. 


CHAPTER   FIVE 


THE  PRINCIPLE  OF  LEAST  ACTION 


1.  RETURN  TO  FERMAT. 

On  January  1st,  1662,  Fermat  wrote  to  C.  de  la  Chambre  concerning 
refraction. 

"  M.  Descartes  has  never  demonstrated  his  principle.  For  apart 
from  the  fact  that  comparisons  are  not  of  much  use  as  foundations  for 
demonstrations,  he  uses  his  own  in  the  wrong  way  and  even  supposes 
that  the  passage  of  light  is  easier  in  dense  bodies  than  in  rare  ones, 
which  is  clearly  false.  " 

In  his  investigation  of  the  refraction  of  light,  Fermat  starts  "  from 
the  principle ,  so  common  and  so  well-established,  that  Nature  always  acts 

in  the  shortest  ways.  " 

He  first  shows  that  in  a  parti 
cular  numerical  example,  the  recti 
linear  path  is  not  the  most  rapid  for 
the  traversal  of  two  media  by  light. 
If  the  medium  AGB  is  supposed 
to  be  more  dense  than  the  medium 
ACB,  "  so  that  the  passage  through 
the  rarer  medium  is  twice  as  easy  as 
that  through  the  denser  one,  "  the 
time  taken  by  the  light  in  going  from 
C  to  G  by  the  straight  line  COG  can 
FiS-  89  be  represented  "  by  the  sum  of  half 

CO  and  the  whole  of  OG.  " 
Taking  CO  =  10,  H 0  =  OD  =  8  and  OF  =  1,  Fermat  shows  that 


+OG-15 


CF  = 


FG  =  ^/85 


CQ 


and  that,  consequently,—-  -f  FG  is  less  than  —  ,  and  therefore  less  than  15. 

2  4 


THE   PRINCIPLE    OF   LEAST   ACTION 


255 


Fermat  adds,  "  I  arrived  at  this  point  without  much  trouble,  but 
it  was  necessary  to  carry  the  investigation  further  ;  and  because,  in 
order  that  my  conscience  might  be  satisfied,  it  was  not  sufficient  to 
have  found  a  point  such  as  F  through  which  the  natural  motion  was 
accomplished  more  quickly,  more  easily  and  in  less  time  than  by  the 
straight  line  COG,  it  was  also  necessary  to  find  the  point  which  allowed 
the  passage  from  one  side  to  the  other  in  less  time  than  any  other  there 
might  be.  In  this  connection,  it  was  necessary  to  use  my  method  of 
maximis  and  minimis,  which  is  rather  successful  for  expediting  this 
kind  of  problem.  " 

Fermat  did  not  doubt  the  truth  of  his  principle,  but  he  had  been 
warned  from  all  sides  that  experiments  confirmed  Descartes'  law. 
Therefore  it  was  dangerous  to  try  to  introduce  a  "  proportion  different 
from  those  which  M.  Descartes  has  given  to  refractions.  '*  Moreover, 
it  was  necessary  to  "  overcome  the  length  and  the  difficulty  of  the  calcu 
lation,  which  at  first  presented  four  lines  by  their  fourth  roots  and 
accordingly  became  entangled  in  assymmetries.  ..."  However,  his 
"  passionate  desire "  to  succeed  fortunately  inspired  him  to  find  a 
method  which  shortened  his  work  by  a  half,  in  reducing  these  four 
asymmetries  to  only  two. 

Fermat's  calculation  is  found  in  his  paper  Synthesis  ad  refractiones, 
probably  written  in  February,  1662. 


Fig.  90 


Let  there  be  a  circle  of  diameter  ANB,  an  incident  ray  MZV  and  a 
refracted  ray  NH.  Let  MRH  be  another  trajectory  passing  through 
any  point  of  AB,  chosen,  for  example,  on  the  right  of  IV. 


256  THE   PRINCIPLES  OF  CLASSICAL  MECHANICS 

Fermat  introduces  the  ratios 

velocity  on  MN  _  MN  __  velocity  on  MR  _  MR 
*  '          velocity  on  NH  ~~  ~IN  ""  velocity  on  RH  ~~  PJR  > 

whence 

time  on  MN  _  MN     _IN_       IN 
time  on  NH  ~  ~N3  '  MN  ~  NH 

and,  similarly, 

time  on  MR       PR 


time  on  RH  " 

so  that 

time  on  MIVH  =  IN  +  NH 
time  on  MRH  ~~  PR  +  RH' 

The  point  jff,  "  at  which  Nature  herself  takes  aim  "  corresponds  to 
a  projection  on  AB  such  that 


= 
v  '  ivs    "ivi 

It  is  necessary  to  show  that 

PR  +  RH  >  IN  +  NH. 
Immediately 

DIV       MR,      ...       ..„. 
-  =  -  by  (1)  and  (2). 
NS       PR       ^  '         V  ; 


Putting 


MN      RN        .     DN      NO 
and 


DJV       NO  NS       NV 

,„    f    As  DJV  <  MN,    therefore     JVO  <  RN 

(  '  \     As  NS  <  DN,     therefore     NV<  NO. 

Now,  by  (3), 

MR*  =  MN2  +  NRZ  +  2DN-NR  =  MN2  +  NR*  +  2MN-NO. 
Therefore,  by  (4) 

(5)  MR  >  MN  +  NO. 

Now,  by  (1),  (2),  (3), 

DN  =  MN=  NO  =  JVO  +  MJV       MR 
NS        IN       NV       NV  +  IN 


THE    PRINCIPLE    OF    LEAST   ACTION  257 

Therefore,  by  (5), 

RP>  NI  +  NV. 

It  remains  to  prove  that 

RH>  HV 

for  then  it  is  clear  that 

PR  +  RH  >  NI  +  NH. 

Now 

RH2  =  NH2  +  NR*  —  2SN-NR 

and  by  (3) 

DN       NO 


DN       JWV       NO 

Therefore 

HN       NR 


NS       NV 


or     SN-NR  =  HN-NV 


since,  by  (4),  NR  >  NV  it  follows  that  RH>NH—NV=HV, 
which  completes  the  proof. 

We  return  to  the  letter  which  we  quoted  at  the  beginning  of  this 
section.  Fermat  concludes,  "  The  reward  of  my  work  has  been  most 
extraordinary,  most  unexpected,  and  the  most  fortunate  that  I  have 
ever  obtained.  For  after  having  gone  through  all  the  equations,  multi 
plications,  antitheses  and  other  operations  of  my  method,  and  finally 
having  settled  the  problem.  .  .  ,  I  found  that  my  principle  gave  exactly 
the  same  proportion  of  the  refractions  that  M.  Descartes  has  established. 
I  was  so  surprised  by  a  happening  that  was  so  little  expected  that  I  only 
recovered  from  my  astonishment  with  difficulty.  I  repeated  my  alge 
braic  operations  several  times  and  the  result  was  always  the  same, 
though  my  demonstration  supposes  that  the  passage  of  light  through 
dense  bodies  is  more  difficult  than  through  rare  ones — something  I 
believe  to  be  very  true  and  necessary,  and  something  which  M.  Descartes 
believes  to  be  the  contrary. 

"  What  must  we  conclude  from  this  ?  Is  it  not  sufficient,  Sir,  that 
as  friends  of  M.  Descartes,  I  might  allow  him  free  possession  of  his 
theorem  ?  Is  it  not  rather  glorious  to  have  learned  the  ways  of  Nature 
in  one  glance,  and  without  the  help  of  any  demonstration  ?  I  therefore 
cede  to  him  the  victory  and  the  field  of  battle.  ..." 


258  THE  PRINCIPLES  OF  CLASSICAL  MECHANICS 

2.  CARTESIAN  OBJECTIONS  TO  FERMAT'S  PRINCIPLE. 

Although  his  demonstration  was  mathematically  incontestible, 
Fermat  was  not  successful  in  convincing  the  Cartesians,  who  opposed 
it  with  metaphysical  objections — which,  at  that  time,  took  place  over 
pure  and  simple  reason. 

These  facts  emerge  from  the  correspondence  between  Fermat  and 
Clerselier.  Thus  Clerselier,  writing  to  Fermat  on  May  6th,  1662, 
declares  that  Fermat's  principle  is,  in  his  eyes,  "  a  principle  which  is 
moral  and  in  no  way  physical  ;  which  is  not,  and  which  cannot  be,  the 
cause  of  any  effect  of  Nature.  "  To  Clerselier,  the  straight  line  is  the 
only  determinate — "  this  is  the  only  thing  that  Nature  tends  to  in  all 
her  motions.  "  And  he  explains — 

"  The  shortness  of  the  time  ?  Never.  For  when  the  radius  MN 
has  come  to  the  point  JV,  according  to  this  principle  it  must  there  be 
indifferent  to  going  to  all  parts  of  the  circumference  BHA,  since  it  takes 
as  much  time  to  travel  to  one  as  to  the  other.  And  since  this  reason  of 
the  shortness  of  time  will  not,  then,  be  able  to  direct  it  towards  one 
place  rather  than  towards  another,  there  will  be  good  reason  that  it 
must  follow  the  straight  line.  For  in  order  that  it  might  select  the 
point  H  rather  than  any  other,  it  is  necessary  to  suppose  that  this  ray 
MN,  which  Nature  cannot  send  out  without  an  indefinite  tendency 
towards  a  straight  line,  remembers  that  it  has  started  from  the  point  M 
with  the  order  to  discover,  at  the  meeting  between  the  two  media, 
the  path  that  it  must  then  travel  in  order  to  arrive  at  H  in  the 
shortest  time.  This  is  certainly  imaginary,  and  in  no  way  founded  on 
physics. 

"  Therefore  what  will  make  the  direction  of  the  ray  MN  (when  it 
has  come  to  IV)  change  at  the  meeting  with  the  other  medium,  if  not 
that  which  M.  Descartes  urges  ?  Which  is  that  the  same  force  that 
acts  on  and  moves  the  ray  MIV,  finding  a  different  natural  arrangement 
for  receiving  its  action  in  this  medium  than  in  the  other,  one  which 
changes  its  own  in  this  respect,  makes  the  direction  of  the  ray  conform 
to  the  disposition  that  it  has  at  the  time.  " 

And  Clerselier  concludes — 

"  That  path,  which  you  reckon  the  shortest  because  it  is  the  quickest, 
is  only  a  path  of  error  and  bewilderment,  which  Nature  in  no  way 
follows  and  cannot  intend  to  follow.  For,  as  Nature  is  determinate 
in  everything  she  does,  she  will  only  and  always  tend  to  conduct  her 
works  in  a  straight  line.  " 

As  for  the  velocity  of  light  in  dense  and  rare  bodies,  Clerselier  believed 
that  it  would  be  "  clearly  more  reasonable  "  to  accept  Fermat's  thesis. 


THE   PRINCIPLE    OF   LEAST   ACTION  259 

But,  with  a  fine  assurance,  lie  writes,  "  M.  Descartes — in  the  23rd  page 
of  his  Dioptrique — proves  and  does  not  simply  suppose,  that  light  moves 
more  easily  through  dense  bodies  than  through  rare  ones.  " 

A  letter  from  Fermat  to  Clerselier,  dated  May  21st,  1662,  contains 
the  following  bitter  ironical  reply. 

"  I  have  often  said  to  M.  de  la  Chambre  and  yourself  that  I  do  not 
claim  and  that  I  have  never  claimed,  to  be  in  the  private  confidence  of 
Nature.  She  has  obscure  and  hidden  ways  that  I  have  never  had  the 
initiative  to  penetrate  ;  I  have  merely  offered  her  a  small  geometrical 
assistance  in  the  matter  of  refraction,  supposing  that  she  has  need 
of  it.  But  since  you,  Sir,  assure  me  that  she  can  conduct  her 
affairs  without  this,  and  that  she  is  satisfied  with  the  order  that 
M.  Descartes  has  prescribed  for  her,  I  willingly  relinquish  my  pre 
tended  conquest  of  physics  and  shall  be  content  if  you  will  leave  me 
with  a  geometrical  problem,  quite  pure  and  in  abstracto,  by  means  of 
which  there  can  be  found  the  path  of  a  particle  which  travels  through 
two  different  media  and  seeks  to  accomplish  its  motion  as  quickly  as  it 
can.  " 

Thus  the  problem  was  taken  back  on  to  the  mathematical  plane, 
the  only  profitable  one. 

In  a  letter  written  in  1664  to  an  unknown  person,  Fermat  returns 
to  "  the  intrigue  of  our  dioptrics  and  our  refractions.  "  If  one  is  to 
judge  from  the  text,  the  Cartesians  had  not  confessed  themselves 
beaten. 

46  The  Cartesian  gentlemen  turned  my  demonstration,  which  was  com 
municated  to  them  by  M.  de  la  Chambre,  upside  down.  At  first  they 
were  of  the  opinion  that  it  must  be  rejected,  and  although  I  represented 
to  them  very  sweetly  that  they  might  be  content  that  the  field  of  battle 
should  remain  with  M.  Descartes,  since  his  opinion  was  justified  and 
confirmed,  albeit  by  reasons  different  from  his  own  ;  that  the  most 
famous  conquerors  did  not  regard  themselves  less  fortunate  when  their 
victory  was  won  with  auxiliary  troops  than  if  it  was  won  by  their  own. 
At  first  they  had  no  wish  to  listen  to  raillery.  They  determined  that 
my  demonstration  was  faulty  because  it  could  not  exist  without  des 
troying  that  of  M.  Descartes,  which  they  always  understood  to  have 
no  equal. .  .  .  Eventually  they  congratulated  me,  by  means  of  a  letter 
from  M.  Clerselier.  .  .  .  They  acclaimed  as  a  miracle  the  fact  that  the 
same  truth  had  been  found  at  the  ends  of  two  such  completely  opposed 
paths  and  announced  that  they  would  prefer  to  leave  the  matter  un 
decided,  saying  that  they  did  not  know,  in  this  connection,  whether  to 
value  M.  Descartes'  demonstration  more  highly  than  my  own,  and 
that  posterity  would  be  the  judge.  " 


260  THE   PRINCIPLES  OF  CLASSICAL  MECHANICS 


In  a  paper  in  the  Ada  of  Leipzig  for  1682,  Leibniz  rejected  Fermat's 
principle.  Light  chooses  the  easiest  path,  which  must  not  be  confused 
with  the  shortest  path  or  with  that  which  takes  the  shortest  time. 

Leibniz  contemplated  a  path  of  least  resistance  or,  more  accurately, 
a  path  for  which  the  product  of  the  path  and  the  "  resistance  "  might 
be  a  minimum.  Leibniz  also  supported  Descartes'  opinion  on  the  relative 
velocity  of  light  in  rare  and  dense  bodies  with  the  aid  of  the  following 
arguments.  Although  glass  "  resists  "  more  than  air,  light  proceeds 
more  quickly  in  glass  than  in  air  because  the  greater  resistance  prevents 
the  diffusion  of  the  rays,  which  are  confined  in  the  passage  after  the 
manner  of  a  river  which  flows  in  a  narrow  bed  and  thus  acquires  a 
greater  velocity. 

4.  MAUPERTUIS'  LAW  OF  REST. 

Before  coming  to  Maupertuis'  dynamics,  we  shall  devote  a  little 
attention  to  a  law  of  mini  mum  and  maximum  which  was  put  forward 
by  this  author  in  the  Memoires  de  VAcademie  des  Sciences  for  1740,  and 
in  which  the  concept  of  potential  makes  its  appearance. 

46  Let  there  be  a  system  of  bodies  which  gravitate,  or  which  are 
attracted  towards  centres  by  the  forces  that  act  on  each  one,  as  the 
7ith  power  of  their  distances  from  the  centre.  In  order  that  all  these 
bodies  should  remain  at  rest,  it  is  necessary  that  the  sum  of  the  products 
of  each  mass  with  the  intensity  of  the  force  1  and  with  the  (n  +  l)th 
power  of  its  distance  from  the  centre  of  its  force  (which  may  be  called 
the  sum  of  the  forces  at  rest)  should  be  a  maximum  or  a  minimum.  " 

By  means  of  this  law  of  rest  Maupertuis  rediscovered  the  essential 
theorems  of  elementary  statics  (the  rule  of  the  parallelogram,  the 
equilibrium  of  an  angular  lever). 

5.  THE  PRINCIPLE   OF  LEAST  ACTION  IN  MAUPERTUIS'  SENSE   (1744). 

The  debate  between  Fermat  and  the  Cartesians,  and  Leibniz's 
objections  to  Fermat's  principle,  prepared  the  way  for  Maupertuis' 
intervention.  The  latter  stated  the  principle  of  least  action  in  a  paper 
read  to  the  Academic  des  Sciences  on  April  15th,  1744.  The  paper  is 
entitled  The  agreement  between  the  different  laws  of  Nature  that  had, 
until  now,  seemed  incompatible. 

1  The  force  is  here  of  the  form  kmrn. 


THE   PRINCIPLE    OF   LEAST   ACTION  261 

Haupertuis  starts  by  recalling  the  laws  which  light  must  obey  — 
rectilinear  propagation  in  a  uniform  medium,  the  law  of  reflection  and 
the  law  of  refraction.  He  seeks  simple  mechanical  analogies. 

"  The  first  of  the  laws  is  common  to  light  and  to  all  bodies.  They 
move  in  a  straight  line  unless  some  outside  force  deflects  them. 

44  The  second  is  also  the  same  as  that  followed  by  an  elastic  ball 
which  is  thrown  at  an  immoveable  surface. 

44  But  it  is  also  very  necessary  that  the  third  law  should  be  explained 
as  satisfactorily.  When  light  passes  from  one  medium  into  another, 
the  phenomena  are  quite  different  from  those  which  occur  when  a  ball 
is  reflected  from  a  surface  which  does  not  yield  to  it  in  any  way  ;  or 
those  which  occur  when  a  ball,  on  meeting  one  that  does  yield  to  it, 
continues  its  progress,  only  changing  the  direction  of  its  path.  .  .  . 
Several  mathematicians  have  extracted  some  fallacy  which  had  escaped 
the  notice  of  Descartes,  and  have  made  the  error  of  his  explanation 
clear. 

"  Newton  gave  up  the  attempt  to  deduce  the  phenomena  of  refrac 
tion  from  those  which  occur  when  a  body  encounters  an  obstacle,  or 
when  it  is  forced  along  in  media  that  resist  differently,  and  fell  back  on 
his  attraction.  Once  this  force,  which  is  distributed  through  all  bodies 
in  proportion  to  the  quantity  of  matter,  is  assumed,  the  phenomena  of 
refraction  are  explained  in  the  most  correct  and  rigorous  way.  .  .  . 

44  M.  Clairaut,  who  assumes  that  light  has  a  tendency  towards 
transparent  bodies,  and  who  considers  this  to  be  caused  by  some  atmo 
sphere  which  could  produce  the  same  effects  as  the  attraction,  has 
deduced  the  phenomena  of  refraction.  .  .  . 

44  Fermat  was  the  first  to  become  aware  of  the  error  of  Descartes' 
explanation.  .  .  .  He  did  not  rely  on  atmospheres  about  the  bodies, 
or  on  attraction,  although  it  is  known  that  the  latter  principle  was 
neither  unknown  nor  disagreeable  to  him.1  He  sought  the  explanation 
of  these  phenomena  in  a  principle  that  was  quite  different  and  purely 
metaphysical. 

44  This  principle  was  4  that  Nature,  in  the  production  of  her  effects, 
always  acts  in  the  most  simple  ways.  '  Therefore  Fermat  believed  that, 


1  MAUPERTUIS  is  here  referring  to  a  passage  from  FERMAT'S  work  (var.  oper. 
p.  114)  and  which  he  cited  elsewhere  with  the  intention  of  showing  that  FERMAT 
had  anticipated  NEWTON.  This  does  not  seem  very  convincing,  for  FERMAT'S  attrac 
tion  remained  metaphysical  in  essence.  Here  is  this  passage.  "  The  common  opinion 
is  that  gravity  is  a  quality  which  resides  in  the  falling  body  itself.  Others  are  of  the 
opinion  that  the  descent  of  bodies  is  due  to  the  attraction  of  another  body,  like  the 
Earth,  which  draws  those  that  descend  towards  itself.  There  is  a  third  possibility  — 
that  it  is  a  mutual  attraction  between  the  bodies  which  is  caused  by  the  mutual 
attraction  that  bodies  have  for  each  other,  as  is  apparent  for  iron  and  a  magnet.  " 


262  THE   PRINCIPLES  OF  CLASSICAL   MECHANICS 

in  all  circumstances,  light  followed  at  once  the  shortest  path  in  the 
shortest  time.1  This  led  him  to  assume  that  light  moved  more  easily 
and  more  quickly  in  the  rarer  media  than  in  those  in  which  there  is  a 
greater  quantity  of  matter.  " 

When  Maupertuis  wrote  it  was  generally  agreed  that  light  moved 
more  quickly  in  denser  media,  in  the  manner  specified  by  the  newtonian 
law  of  the  proportionality  of  the  indices  of  refraction  to  the  velocities 
of  propagation. 

"All  the  structure  that  Fermat  has  built  up  is  therefore  destroyed 

In  the  paper  that  M.  de  Mayran  has  given  on  the  reflection  and  re 
fraction,  there  can  be  found  the  history  of  the  dispute  between  Fermat 
and  Descartes,  and  the  difficulty  and  inability  there  has  so  far  been  to 
reconcile  the  law  of  refraction  with  the  metaphysical  principle.  " 

Therefore,  unlike  Fermat,  Maupertuis  sought  a  minimum  principle 
that  might  be  compatible  with  the  newtonian  law,  and  not  with  the 
now  generally  accepted  law  which  goes  back  to  Huyghens.  The  strange 
thing  is  not  that  he  succeeded  in  finding  it.  Rather  it  is  that,  in  boldly 
extending — one  is  even  tempted  to  say  gratuitously  extending — this 
minimum  principle  into  the  field  of  dynamics,  he  was  led  to  a  law  which 
was  truly  sufficient,  and  which  he  successfully  opposed  to  the  thesis  of 
Descartes  on  the  conservation  of  momentum  and  of  Leibniz  on  the  con 
servation  of  kinetic  energy. 

Up  to  this  point,  our  author  has  only  criticised  the  different  inter 
pretations  of  the  laws  of  refraction  that  had  been  put  forward.  We 
shall  now  look  at  his  achievement.  The  relevant  passage  merits  quota 
tion  in  its  entirely — on  the  rational  plane,  it  would  be  impossible  to 
conceal  its  extreme  weakness. 

We  now  enter  the  metaphysical  plane  in  the  most  complete  sense 
of  the  word. 

"  In  meditating  deeply  on  this  matter,  I  thought  that,  since  light 
has  already  forsaken  the  shortest  path  when  it  goes  from  one  medium 
to  another — the  path  which  is  a  straight  line — it  could  just  as  well  not 
follow  that  of  the  shortest  time.  Indeed,  what  preference  can  there 
be  in  this  matter  for  time  or  distance  ?  Light  cannot  at  once  travel 
along  the  shortest  path  and  along  that  of  the  shortest  time — why  should 
it  go  by  one  of  these  paths  rather  than  by  the  other  ?  Further,  why 
should  it  follow  either  of  these  two  ?  It  chooses  a  path  which  has  a 
very  real  advantage —  the  path  which  it  takes  is  that  by  which  the  quantity 
of  action  is  the  least. 

1  As  far  as  it  concerns  the  path,  this  is  incorrect.  What  is  a  minimum,  to  FERMAT, 
is  the  sum  In  -f  I'n',  the  sum  of  the  products  of  each  trajectory  with  the  corresponding 
refractive  index  in  SNELL'S  sense. 


THE    PRINCIPLE    OF   LEAST   ACTION  263 

"  It  must  now  be  explained  what  I  mean  by  the  quantity  of  action. 
When  a  body  is  carried  from  one  point  to  another  a  certain  action  is 
necessary.  This  action  depends  on  the  velocity  that  the  body  has  and 
the  distance  that  it  travels,  but  it  is  neither  the  velocity  nor  the  distance 
taken  separately.  The  quantity  of  action  is  the  greater  as  the  velocity 
is  the  greater  and  the  path  which  it  travels  is  the  longer.  It  is  propor 
tional  to  the  sum  of  the  distances,  each  one  multiplied  by  the  velocity 
with  which  the  body  travels  along  it.1 

"It  is  the  quantity  of  action  which  is  Nature's  true  storehouse ,  and 
which  it  economises  as  much  as  possible  in  the  motion  of  light.  " 

Maupertuis'  demonstration  follows. 

"  Let  there  be  two  different  media,  separated  by  a  surface  which  is 
represented  by  the  line  CD,  such  that  the  velocity  in  the  upper  medium 
is  proportional  to  m  and  the  velocity  in  the  lower  medium  is  propor 
tional  to  n.  Let  there  be  a  ray  of  light,  starting  from  the  given  point  A, 
which  must  pass  through  the  given  point  B.  In  order  to  find  the  point 
JR  at  which  it  must  break  through,  I  seek  the  point  at  which,  if  the  ray 
breaks  through,  the  quantity  of  action  is  least.  I  have  m  •  AR  +  n>  •  RB, 
which  must  be  a  minimum. 

66  Or,  having  drawn  the  perpendiculars  AC,  BD,  to  the  common 
surface  of  the  two  media,  I  have 


m 


VAC*  +  CR*  +  n  VBD*  +  DR*  = 


ff  D 


C 


Fig.  91 
or,  since  AC  and  BP  are  constants 


nPR-dDR     __ 


VAC*  +  CR2     VBP*  + 


1  A  footnote  adds  the  following  detail —  "  As  there  is  only  one  body,  the  mass  is 
neglected.  " 


264  THE   PRINCIPLES   OF  CLASSICAL   MECHANICS 

"  But,  since  CD  is  a  constant,  there  obtains 

dCR  =  —  dDR. 

"  Therefore 

mCR       nDR         ,      CR     RD  .  .  n 
and 


AR         BR  AR     BR       m 

or,  in  words,  the  sine  of  the  incidence,  or  the  sine  of  the  refraction,  are  in 
inverse  proportion  to  the  velocity  which  the  light  has  in  each  medium. 

"  All  the  phenomena  of  refraction  now  agree  with  the  great  principle 
that  Nature  in  the  production  of  her  works,  always  acts  in  the  most  simple 
ways.  " 

Maupertuis  then  shows  without  difficulty  that  "  this  basis,  this 
quantity  of  action  that  nature  economises  in  the  motion  of  light  through 
different  media,  she  also  saves  in  the  reflection  and  the  linear  propa 
gation.  In  both  these  circumstances,  the  least  action  reduces  to  the 
shortest  path  and  the  shortest  time.  And  it  is  this  consequence  that 
Fermat  took  as  a  principle.  " 

Maupertuis  concludes,  "  I  know  of  the  repugnance  that  several 
mathematicians  have  for  final  causes  when  applied  to  physics,  and  to 
a  certain  extent  I  am  in  accord  with  them.  I  believe  that  they  are  not 
introduced  without  risk.  The  error,  which  men  like  Fermat  and  those 
that  followed  him,  have  committed,  only  shows  that,  too  often,  their 
use  is  dangerous.  It  can  be  said,  however,  that  it  is  not  the  principle 
which  has  betrayed  them,  but  rather,  the  haste  with  which  they  have 
taken  for  the  principle  what  is  merely  one  of  the  consequences  of  it. 
It  cannot  be  doubted  that  all  things  are  regulated  by  a  Supreme  Being 
who,  when  he  impressed  on  matter  the  forces  which  denote  his  power, 
destined  it  to  effect  the  doings  which  indicate  his  wisdom.  " 

6.   THE  APPLICATION  OF  THE  PRINCIPLE  OF  LEAST  ACTION  TO  THE  DIRECT 
IMPACT  OF  TWO  BODIES. 

In  a  paper  published  by  the  Royal  Academy  of  Berlin  in  1747,  and 
called  On  the  laws  of  motion  and  of  rest,  Maupertuis  applied  the  principle 
of  least  action  to  the  direct  impact  of  two  bodies. 

He  only  considered  the  effect  of  the  direct  impact  of  two  homogeneous 
spheres,  and  started  from  the  hypothesis  that  "  the  magnitude  of  the 
impact  of  two  given  bodies  depends  uniquely  on  their  respective  velocity,  " 
that  is,  on  their  relative  velocity.  He  distinguished  between — 

"  Perfectly  hard  bodies.  These  are  those  whose  parts  are  inseparable 
and  inflexible,  and  whose  shape  is  consequently  unalterable. 


THE   PRINCIPLE    OF    LEAST   ACTION  265 

"  Perfectly  elastic  bodies.  These  are  those  whose  parts,  after  being 
deformed,  right  themselves  again,  taking  up  their  original  situation  and 
restoring  to  the  body  its  original  shape.  " 

Modern  language  would  call  the  first  category  completely  devoid  of 
elasticity  or  perfectly  soft.  But  the  important  matter  here  is  that  of  the 
experimental  laws  which  Maupertuis  stated. 

"After  the  impact,  hard  bodies  travel  together  with  a  common  velocity.  . . . 
The  respective  velocity  of  elastic  bodies  after  the  impact  is  the  same  as  that 
before.  " 

Maupertuis  did  not  treat  the  intermediate  case,  "  that  of  soft  or 
fluid  bodies,  which  are  merely  aggregates  of  hard  or  elastic  ones.  " 

He  started  from  the  principle  that  "  when  any  change  takes  place 
in  Nature,  the  quantity  of  action  necessary  for  this  change  is  the  smallest 
possible.  " 

We  shall  quote  (notation  apart)  Maupertuis'  argument  on  the  impact 
of  hard  bodies. 

"  Let  there  be  two  hard  bodies  A  and  B,  whose  masses  are  m  and 
/ra',  which  move  in  the  same  direction  with  velocities  v  and  v0  ;  but  A 
more  quickly  than  B,  so  that  it  overtakes  B  and  collides  with  it.  Let 
the  common  velocity  of  the  two  bodies  after  the  impact  =  vx  <  v0 
and  >  VQ.  The  change  which  occurs  in  the  Universe  consists  in  that 
the  body  A,  which  used  to  move  with  a  velocity  VQ  and  which,  in  a 
certain  time,  used  to  travel  a  distance  =  t?0,  now  moves  with  the  velocity 
i?!  and  travels  no  more  than  a  distance  =  v±.  The  body  -B,  which  only 
used  to  move  with  a  velocity  v'0  and  travelled  a  distance  =  v^  moves 
with  the  velocity  vl  and  travels  a  distance  =  v±. 

"  This  change  is  therefore  the  same  as  would  occur  if,  while  the  body 
A  moved  with  the  velocity  v0  and  travelled  a  distance  =  t;0,  it  were 
carried  backwards  on  an  immaterial  plane,  which  was  made  to  move 
with  a  velocity  t;0 — v^  through  a  distance  =  t;0 — i7x  ;  and  that  while 
the  body  B  moved  with  the  velocity  v$  and  travelled  a  distance  =  v& 
it  were  carried  forwards  by  an  immaterial  plane,  which  was  made  to 
move  with  a  velocity  v± — VQ  through  a  distance  v± — VQ. 

"  Now  whether  the  bodies  A  and  B  move  with  their  appropriate 
velocities  on  the  moving  planes,  or  whether  they  are  immobile  there, 
the  motion  of  the  planes  loaded  with  these  bodies  being  the  same,  the 
quantities  of  action  produced  in  Nature  will  be  m(vQ  —  t;x)2  and 
m/(vi  —  vo)2'  an<^  fr  *s  the  sum  °£  these  which  must  be  as  small  as 
possible.  Therefore  it  must  be  that 

mv\  —  2mvQv1  +  mv\  +  Hz't;f  —  2mfv1VQ  +  mrv'^  =  Min. 
or  —  2mv0dv1  -\-  2mv1dvl  +  2mfvidvl  —  ^m'v^dv^  =  0 


266  THE   PRINCIPLES  OF  CLASSICAL   MECHANICS 

whence  the  common  velocity 

mvn  +  mfv,( 


1  m  +  m' 

is  obtained.  " 

No  purpose  would  be  served  by  reproducing  the  argument  relevant 
to  two  bodies  moving  towards  each  other.  Here  too  the  condition  of 
least  action  reduces  to  the  conservation  of  the  total  momentum. 

Next  treating  the  impact  of  elastic  bodies,  Maupertuis  used  an 
argument  which  was  completely  analogous  to  that  which  we  have 
reproduced.  Apart  from  sign,  the  "  respective  "  velocity  is  conserved 
after  the  impact,  or 


The  cpiantity  of  action  involved  has  the  value 


and  it  follows  from  the  condition  of  least  action  that 


_  __ 

1  ~~  m  +  m  1  m  +  m' 

On  this  occasion  the  living  forces  are  conserved,  "  but  this  con 
servation  only  takes  place  for  elastic  bodies,  not  for  hard  ones.  The 
general  principle,  which  applies  to  the  first  and  to  the  others,  is  that  the 
quantity  of  action  necessary  to  produce  some  change  in  Nature  is  the 
smallest  that  is  possible.  " 

At  the  end  of  his  paper,  Maupertuis  dealt  with  the  principle  of  the 
lever,  and  deduced  it  from  the  principle  of  least  action. 

"  Let  c  be  the  length  of  the  lever,  which  I  suppose  to  be  immaterial, 
and  at  whose  ends  are  placed  two  bodies  whose  masses  are  A  and  B. 
Let  z  be  the  distance  of  the  body  A  from  the  point  of  support  which  is 
sought,  and  c  —  %  be  the  distance  of  the  body  B.  It  is  clear  that,  if  the 
lever  has  some  small  motion,  the  bodies  A  and  B  will  describe  small 
arcs  which  are  similar  to  each  other  and  proportional  to  the  distances 
of  the  bodies  from  the  point  which  is  sought.  Therefore  these  arcs  will 
be  the  distances  travelled  by  the  bodies,  and  at  the  same  time  will 
represent  their  velocities.  The  quantity  of  action  will  therefore  be 
proportional  to  the  product  of  each  body  by  the  square  of  its  arc.  Or 
(since  the  arcs  are  similar)  to  the  product  of  each  body  by  the  square 
of  its  distance  from  the  point  about  which  the  lever  turns,  that  is,  to 


THE   PRINCIPLE    OF    LEAST   ACTION  267 

Az~  and  B(c  —  s)2,  and  it  is  the  sum  of  these  which  must  be  the  smallest 
possible.     Therefore 

Az*+  B(c-z)*  =  Min. 

or 

2Azdz  +  2Bzdz  —  ZBcdz  =  0 

from  which  it  is  deduced  that 

Be 


Z~~ 


A  +  B 
which  is  the  fundamental  proposition  of  statics.  " 

7.    THE  PRINCIPLE   OF  LEAST  ACTION  IN  MAUPERTUIS'   WORK. 

Maupertuis,  who  had  been  a  musketeer,  had  a  great  liking  for  geo 
metry.  He  was  a  surveyor  and,  in  an  amateur  way,  a  geographer, 
astronomer,  biologist,  moralist  and  linguist.  .  .  .  And  to  crown  and 
grace  it  all,  Maupertuis  was  a  metaphysician.  Although  he  had  a 
systematic  mind,  because  of  a  trait  rather  common  to  men  of  his  pro 
vince  he  was  not  free  from  fantasy.  From  this  fantasy,  or  perhaps 
from  his  temperament,  sprang  naivety. 

We  shall  therefore  turn  over  the  pages  of  Maupertuis'  work,  seeking 
an  explanation  of  the  principle  of  least  action.1 

Here  we  shall  only  dwell  on  the  Essai  de  Cosmologie.  In  this  docu 
ment  Maupertuis  contrasted  the  rationalist  school,  "  wishing  to  submit 
Nature  to  a  purely  material  regime  and  to  ban  final  causes  entirely,  " 
with  the  school  which,  on  the  contrary,  "  makes  continual  use  of  these 
causes  and  discovers  the  intentions  of  the  Creator  in  every  part  of 
Nature.  .  .  .  According  to  the  first,  the  Universe  could  dispense  with 
God  ;  according  to  the  second,  the  tiniest  parts  of  the  Universe  are  as 
much  demonstrations  "  of  the  existence  of  God. 

He  declared,  "  I  have  been  attacked  by  both  these  factions  of 
philosophy.  .  .  .  Reason  defends  me  from  the  first,  an  enlightened 
century  has  not  allowed  the  other  to  oppress  me.  " 

Thus  Maupertuis  flattered  himself  with  having  found  a  happy  mean 
between  these  two  extreme  attitudes.  "  Those  who  make  immoderate 
use  (of  final  causes)  have  wished  to  persuade  me  that  I  seek  to  deny  the 
evidence  of  the  existence  of  God  —  which  the  Universe  everywhere 
presents  to  the  eyes  of  all  men  —  in  order  to  substitute  for  it  one  which 
has  only  been  given  to  a  few.  " 

1  We  have  referred  both  to  the  Dresden  edition  (Walter,  1752)  and  the  Lyons 
edition  (Bruyset,  1756). 


268  THE    PRINCIPLES   OF  CLASSICAL  MECHANICS 

Among  the  evidence  of  the  existence  of  God,  Maupertuis  intended 
to  dispense  with  all  that  was  provided  by  metaphysics.  He  also  took 
no  account  of  that  which  sprang  from  the  structure  of  animals  and 
plants,  such  as  the  proof— to  cite  only  one — offered  by  the  folds  in  the 
skin  of  a  rhinoceros,  who  would  not  be  able  to  move  without  them. 

"  Philosophers  who  have  assigned  the  cause  of  motion  to  God  have 
been  reduced  to  this  because  they  did  not  know  where  else  to  place  it. 
Not  being  able  to  conceive  that  matter  had  any  ability  to  produce, 
distribute  and  destroy  motion,  they  have  resorted  to  an  Immaterial 
Being.  But  when  it  is  known  that  all  the  laws  of  motion  are  based  on 
the  principle  of  better,  it  cannot  be  doubted  that  these  have  their  found 
ation  in  an  omnipotent  and  omniscient  Being,  whether  he  gave  bodies 
the  power  to  interact  with  each  other,  or  whether  he  used  some  other 
means  which  is  still  less  understood  by  us.  " 

He  was  not  concerned,  like  Fermat,  with  assuming  that  Nature  acts 
in  the  most  simple  ways.  He  was  not  concerned,  as  Descartes  was, 
with  assuming  that  the  same  quantity  of  motion  was  always  conserved 
in  Nature —  "  He  deduced  his  laws  of  motion  from  this  ;  observation 
belied  them,  for  the  principle  was  not  true.  "  Finally  he  was  not 
concerned,  like  Leibniz,  in  assuming  that  the  living  force  was  always 
conserved.  Huyghens  and  Wren  had  discovered  the  laws  of  the  impact 
of  elastic  bodies  simultaneously,  but  Huyghens  had  not  taken  these 
laws  onto  the  plane  of  a  universal  principle.  The  conservation  of  living 
forces  does  not  apply  to  the  impact  of  hard  bodies  and,  on  this  occasion, 
Maupertuis  accused  the  followers  of  Leibniz  "  of  preferring  to  say  that 
there  are  no  hard  bodies  in  Nature  "  than  to  give  up  their  principle. 
"  This  has  been  reduced  to  the  strangest  paradox  that  love  of  a  system 
could  produce —  for  the  primitive  bodies,  the  bodies  that  are  the 
elements  of  all  others,  what  can  they  be  but  hard  bodies  ?  " 

Therefore  Maupertuis  denied  all  general  principles  that  were  not 
final.  "  In  vain  did  Descartes  imagine  a  world  which  could  arise  from 
the  hand  of  the  Creator.  (Strictly  speaking,  Descartes'  system  supposes 
the  initial  intervention  of  the  Creator,  and  the  continuance  of  his 
"  customary  assistance.  ")  In  vain  did  Leibniz,  on  another  principle, 
devise  the  same  plan.  " 

And  he  concludes,  "  After  so  many  great  men  have  worked  on  this 
matter,  I  hardly  dare  say  that  I  have  discovered  the  principle  on  which 
all  the  laws  of  motion  are  founded ;  a  principle  which  applies  equally 
to  hard  bodies  and  elastic  bodies;  from  which  the  motions  of  all  corpo 
real  substances  follow.  .  .  .  Our  principle,  more  in  conformity  with  the 
ideas  of  things  that  we  should  have,  leaves  the  world  in  its  natural 
need  of  the  power  of  the  Creator,  and  is  a  necessary  result  of  the  wisest 


THE    PRINCIPLE    OF    LEAST    ACTION  269 

doing  of  that  same  power.  .  .  .  What  satisfaction  for  the  human  mind, 
in  contemplating  these  laws — so  beautiful  and  so  simple — that  they 
may  be  the  only  ones  that  the  Creator  and  the  Director  of  things  has 
established  in  matter  in  order  to  accomplish  all  the  phenomena  of  the 
visible  world.  " 


8.  D'ALEMBERT'S  CONDEMNATION  OF  FINAL  CAUSES. 

D'Alembert  himself  was  not  directly  involved  in  the  polemic  on  the 
principle  of  least  action  that  we  shall  describe  in  the  next  section.  But 
he  did  completely  condemn  the  intervention  of  final  causes  in  the  prin 
ciples  of  mechanics. 

Indeed,  he  wrote,1  "  The  laws  of  equilibrium  and  of  motion  are 
necessary  truths.  A  metaphysician  would  perhaps  be  satisfied  to  prove 
this  by  saying  that  it  was  the  wisdom  of  the  Creator  and  the  simplicity 
of  his  intentions  never  to  establish  other  laws  of  equilibrium  and  of 
motion  than  those  which  follow  from  the  very  existence  of  bodies  and 
their  mutual  impenetrability.  But  we  have  considered  it  our  duty  to 
abstain  from  this  kind  of  argument,  because  it  has  seemed  to  us  that 
it  is  based  on  too  vague  a  principle.  The  nature  of  the  Supreme  Being 
is  too  well  concealed  for  us  to  be  able  to  know  directly  what  is,  or  is 
not,  in  conformity  with  his  wisdom.2  We  can  only  discover  the  effect 
of  his  wisdom  by  the  observation  of  the  laws  of  nature,  since  mathema 
tical  reasoning  has  made  the  simplicity  of  these  laws  evident  to  us,  and 
experiment  has  shown  us  their  application  and  their  scope. 

"  It  seems  to  me  that  this  consideration  can  be  used  to  judge  the 
value  of  the  demonstrations  of  the  laws  of  motion  which  have  been 
given  by  several  philosophers,  in  accordance  with  the  principle  of  final 
causes  ;  that  is,  according  to  the  intentions  that  the  Author  of  nature 
might  have  formulated  in  establishing  these  laws.  Such  demonstra 
tions  cannot  have  as  much  force  as  those  which  are  preceded  and 
supported  by  direct  demonstrations,  and  which  are  deduced  from  prin 
ciples  that  are  more  within  our  grasp.  Otherwise,  it  often  happens 
that  they  lead  us  into  error.  It  is  because  he  followed  this  method, 
and  because  he  believed  that  it  was  the  Creator's  wisdom  to  conserve 
the  same  quantity  of  motion  in  the  Universe  always,  that  Descartes 
has  been  misled  about  the  laws  of  impact.3  Those  who  imitate  him 

1  Traite  de  Dynamique,  Discours  preliminaire,  1758  edition,  p.  29. 

2  Clearly  an  allusion  to  MAUPERTUIS. 

3  The  reader  knows  that  DESCARTES'  error  is  not,  in  fact,  that  of  having  asserted 
the  conservation  of  momentum,  but  of  having  considered  m\v\  instead  of  mv. 


270  THE   PRINCIPLES  OF  CLASSICAL   MECHANICS 

run  the  risk  of  being  similarly  deceived  ;  or  of  giving  as  a  principle, 
something  that  is  only  true  in  certain  circumstances  ;  or  finally,  of 
regarding  something  which  is  only  a  mathematical  consequence  of  cer 
tain  formulae  as  a  fundamental  law  of  nature.  " 

9.  THE  POLEMIC  ON  THE  PRINCIPLE  OF  LEAST  ACTION. 

In  the  Acta  of  Leipzig  for  1751  Koenig,  Professor  at  The  Hague, 
reproduced  part  of  a  letter  which  he  aUeged  had  been  written  by  Leibniz 
to  Herman  in  1707,  and  which  contained  the  following  passage. 

"Force  is  therefore  as  the  product  of  the  mass  and  the  square  of 
the  velocity,  and  the  time  plays  no  part,  as  the  demonstration  which 
you  use  shows  clearly.  But  action  is  in  no  way  what  you  think.  There 
the  consideration  of  the  time  enters  as  the  product  of  the  mass  by  the 
distance  and  the  velocity,  or  of  the  time  by  the  living  force.  I  have 
pointed  out  that  in  the  variations  of  motions,  it  usually  becomes  a 
minimum  or  a  maximum.  From  this  can  be  deduced  several  important 
propositions.  It  can  be  used  to  determine  the  curves  described  by 
bodies  that  are  attached  to  one  or  several  centres.  I  wished  to  treat 
these  things  in  the  second  part  of  my  Dynamics  but  I  suppressed  them, 
because  the  hostile  reception  with  which  prejudice,  from  the  first, 
accorded  them,  disgusted  me.  " 

Maupertuis,  for  his  part,  represented  the  affair  in  the  following  way.1 

"  Koenig,  Professor  at  The  Hague,  took  it  into  his  head  to  insert  in 
the  proceedings  of  Leipzig  a  dissertation  in  which  he  had  two  ends  in 
view — rather  contradictory  ones  for  such  a  zealous  partisan  of  M.  de 
Leibniz,  but  which  he  found  it  possible  to  unite.  .  .  .  He  attacked  my 
principle  as  strongly  as  possible.  And,  for  those  that  he  was  unable  to 
persuade  of  its  falsehood,  he  quoted  a  fragment  of  a  letter  from  Leibniz 
from  which  it  could  be  inferred  that  the  principle  belonged  to  that  one.  " 

Summoned  by  Maupertuis  to  produce  the  letter,  Koenig  referred 
him  to  "  a  man  whose  head  has  been  cut  off"  (Henzi,  of  Berne).  No 
trace  of  this  letter  was  found  in  spite  of  all  the  searches  ordered  by  the 
King  at  the  request  of  the  Academie.  The  matter  became  a  very  acri 
monious  one.  "  It  was  no  longer  a  matter  of  reasons.  M.  Koenig  and 
his  supporters  only  replied  with  abuse.  Finally  they  resorted  to  libel. . . ." 

At  the  time,  Maupertuis  presided  over  the  Academy  of  Berlin  on  the 
appointment  of  Frederic  II.  Koenig  returned  his  diploma  to  the 
Academy  and  published  an  Appeal  to  the  public  from  the  judgement 

1  (Euvres  completes,  1756  edition,  Letter  XI.  —  Sur  ce  qui  s'est  passe  d  /'occasion 
du  principe  de  moindre  action. 


THE    PRINCIPLE    OF   LEAST   ACTION  271 

that  the  Academy  of  Berlin  had  pronounced  in  this  matter.  In  1753 
he  emphasised  this  with  a  Defence  of  the  Appeal  to  the  public  which  he 
addressed  to  Maupertuis  and  which  he  claimed  not  only  the  priority 
of  Leibniz,  but  also  that  of  Malebranche,  Wolf,  s'Gravesande  and  Engel- 
hardt. 

Voltaire  took  part  in  the  controversy.  Maupertuis  wrote,1  "  The 
strangest  thing  was  to  see  appear  as  an  auxiliary  in  this  dispute  a  man 
who  had  no  claim  to  take  part.  Not  satisfied  with  deciding  at  random 
on  this  matter — which  demanded  much  knowledge  which  he  lacked — he 
took  this  opportunity  to  hurl  the  grossest  insults  at  me,  and  was  soon 
to  cap  them  with  his  Diatribe.2  I  allowed  this  torrent  of  gall  and  filth 
to  run  on,  when  I  saw  myself  defended  by  the  pen  and  the  sceptre. 
Although  the  most  eloquent  pen  of  all  had  uttered  these  libels,  justice 
made  his  work  burn  on  the  gibbets  and  in  the  public  places  of  Berlin.  " 

"  My  only  fault,  "  declared  Maupertuis,  "  was  that  of  having  disco 
vered  a  principle  that  created  something  of  a  sensation.  "  Euler, 
director  of  the  Academy  of  Berlin,  presented  the  following  report. 
"  This  great  geometer  has  not  only  established  the  principle  more 
firmly  than  I  had  done  but  his  method,  more  ubiquitous  and  penetrating 
than  mine,  has  discovered  consequences  that  I  had  not  obtained.  After 
so  many  vested  interests  in  the  principle  itself,  he  has  shown,  with 
the  same  evidence,  that  I  was  the  only  one  to  whom  the  discovery  could 
be  attributed.  " 


1  Ibid. 

2  La  Diatribe  du  Dr  Akdkia^  medecin  du  Pope,  is  too  well  known  to  need  emphasis 
here.     We  confine  ourselves  to  the  extraction  of  what  is  directly  relevant  to  our  subject. 

At  the  beginning  VOLTAIRE  writes, "  We  ask  forgiveness  of  God  for  baving  pretended 
tbat  tbere  is  only  proof  of  bis  existence  in  A  -f-  B  divided  by  Z,  etc.  ..."  This  is  both 
a  reference  to  the  demonstration  of  the  equilibrium  of  the  lever  by  means  of  the  principle 
of  least  action  and  to  MAUPERTUIS'  rejection,  in  his  Essai  de  Cosmologie,  of  metaphysical 
proofs  of  the  existence  of  God. 

Then,  in  the  guise  of  SL  Decision  of  the  professors  of  the  College  of  Wisdom^  VOLTAIRE 
makes,  in  spite  of  the  malicious  terms  in  which  it  is  couched,  an  accurate  criticism. 
"  The  assertion  that  the  product  of  the  distance  and  the  velocity  is  always  a  minimum 
seems  to  us  to  be  false,  for  this  product  is  sometimes  a  maximum,  as  Leibniz  believed 
and  as  he  has  shown.  It  seems  that  the  young  author  has  only  taken  half  of  Leibniz's 
idea ;  and,  in  this,  we  vindicate  him  of  ever  having  had  an  idea  of  Leibniz  in  its  entirety." 

And  finally,  concerning  the  part  played  by  EULER,  which  MAUPERTUIS  had  not 
thought  of  concealing,  the  same  Decision  declares,  "  We  say  that  the  Copernicus's, 
the  Kepler's,  the  Leibniz's  .  .  .  are  something,  and  that  we  have  studied  under  the 
Bernoulli's,  and  shall  study  again ;  and  that,  finally,  Professor  Euler,  who  was  very 
anxious  to  serve  us  as  a  lieutenant,  is  a  very  great  geometer  who  has  supported  our 
principle  with  formulae  which  we  have  been  quite  unable  to  understand,  but  which 
those  who  do  understand  have  assured  us  they  are  full  of  genius,  like  the  published 
works  of  the  professor  referred  to,  our  lieutenant.  ..." 

We  must  also  add  that  MAUPERTUIS  is  caricatured  in  a  consistently  malicious  way 
in  Microm6ga$,  Candide,  and  in  ISHomme  aux  quarante  ecus. 


272  THE  PRINCIPLES  OF  CLASSICAL  MECHANICS 

10.  EULER'S  JUDGEMENT  ON  THE  CONTROVERSY  ON  LEAST  ACTION. 

Traces  of  Eider's  opinion  about  the  controversy  on  least  action  can 
be  found  in  a  Dissertation  on  the  principle  of  least  action,  with  an  exam 
ination  of  the  objections  to  this  principle  made  by  Professor  Koenig.  This 
was  printed  at  Berlin,  in  Latin  and  French,  in  1753. 

Euler  discloses  a  great  respect  for  Maupertuis,  our  "  illustrious 
President.  "  He  pays  homage  to  Maupertuis  law  of  rest  in  the  following 
terms.  This  principle  indicates  "  the  marvellous  accord  of  the  equili 
brium  of  bodies,  whether  rigid,  flexible,  elastic  or  fluid.  From  each 
attraction  can  be  deduced  the  Efficacy  of  each  force,  and  there  is  equili 
brium  when  the  sum  of  all  the  efficacies  is  least.  " 

Euler  remarks,  "  Professor  Koenig  places  us  under  the  twin  obli 
gation  of  proving  that  the  principle  of  least  action  is  true,  and  that  it 
does  not  belong  to  Leibniz.  " 

To  Koenig,  all  instances  of  equilibrium  can  be  deduced  successfully 
from  the  principle  of  living  forces. 

The  "  Koenigian  principle  "  consists  of  "  the  annihilation  of  the 
living  force  if  there  were  no  equilibrium.  " 

It  can  be  seen,  "  more  clearly  than  the  day,  "  that  where  the  applied 
forces  produce  no  living  force,  there  is  equilibrium.  In  short,  in  stating 
the  principle  of  the  nullity  of  the  living  force  Professor  Koenig  is  "  con 
cealing  that  which  he  found  first,  6  that  in  the  state  of  equilibrium  there 
is  neither  motion  nor  living  force  '.  "  In  this  form,  the  principle  of 
Koenig  may  appear  a  truism  but,  to  be  accurate,  his  method  proceeds 
in  the  following  way.  First,  the  system  is  displaced  from  its  equili 
brium  position  and  the  living  force  calculated.  Then  this  is  cancelled 
out  and  the  conditions  of  equilibrium  deduced.  This  method  searches 
the  difficulty,  for  the  calculation  of  motion  is,  in  general,  more  difficult 
than  that  of  equilibrium.  And  Euler  concludes,  "  Koenig's  principle 
usually  leads  to  great  circumlocutions  and  is,  often,  incapable  of  appli 
cation.  " 

To  Koenig,  action  does  not  differ  from  living  force.  He  considers 
himself  able  to  assert  that  "  It  is  clearly  seen  that  all  equilibrium  arises 
from  the  nullity  of  the  living  force  or  from  the  nullity  of  the  action, 
taken  correctly,  and  in  no  way  from  their  Min.  of  Max.  "  Euler 
forthrightly  condemned  this  thesis  and,  in  passing,  made  the  following 
observation. 

"  Professor  Koenig  seems  too  attached  to  metaphysical  speculations 
to  be  able  successfully  to  withdraw  his  mind  from  those  subtle  abstrac 
tions  and  to  apply  it  to  the  ordinary  and  material  ideas  such  as  those 
which  are  the  subject  of  mechanics.  " 


THE   PRINCIPLE    OF   LEAST   ACTION  273 

In  the  next  section  we  shall  study  Euler's  personal  contribution  to 
the  extremum  principle  in  dynamics.  In  the  document  which  concerns 
us  here,  he  only  made  the  following  allusion  to  this  matter. 

44 1  am  not  in  any  way  concerned,  here,  with  the  observation  which 
I  have  made  in  the  motion  of  the  celestial  bodies  and,  general,  of  those 
attracted  to  fixed  centres  of  force,  that  if  the  mass  of  the  body  is  multi 
plied  by  the  distance  travelled,  at  each  instant,  and  by  the  velocity, 
then  the  sum  of  all  these  products  is  always  the  least.  "  To  Euler,  the 
question  is  one  of  an  a  posteriori  verification  and  not  of  an  a  priori 
deduction. 

Further,  Euler  acknowledged  Maupertuis'  priority  in  the  principle 
of  least  action.  "  Since  this  remark  was  only  made  after  M.  Maupertuis 
had  presented  his  principle,  it  should  not  imply  any  prejudice  against 
his  originality.  " 

11.  EULER  AND  THE  LAW  OF  THE  EXTREMUM  OF  /  mvds. 

As  early  as  1744  Euler  published  a  work  called  Methodus  inveniendi 
lineas  curvas  maximi  minimive  proprietate  gaudentes  (Bousquet,  Lau 
sanne).  Here  we  are  concerned  with  his  Appendix  II —  De  motu 
projectorum  in  medio  non  resistente  per  methodum  maximorum  ac  mini- 
moTum  determinando. 

Euler  starts  from  the  following  principle.  "  Since  all  the  effects  of 
Nature  obey  some  law  of  maximum  or  minimum,  it  cannot  be  denied 
that  the  curves  described  by  projectiles  tinder  the  influence  of  some 
forces  will  enjoy  the  same  property  of  maximum  or  minimum.  It 
seems  less  easy  to  define,  a  priori,  using  metaphysical  principles,  what 
this  property  is.  But  since,  with  the  necessary  application,  it  is  possible 
to  determine  these  curves  by  the  direct  method,  it  may  be  decided  which 
is  a  maximum  or  a  minimum.  "  x 

Euler  emphasised,  in  the  Dissertation  which  we  have  analysed  in  §  10, 
that  the  matter  was,  to  him,  one  of  the  a  posteriori  verification  of  the 
existence  of  an  extremum  in  particular  examples  of  the  dynamics  of  a 
particle. 

The  quantity  which  Euler  considered  was,  at  first,  Mds-\/~v.  Here 
M  is  the  mass  of  the  particle,  ds  the  element  of  distance  travelled  and 
v  the  height  of  fall.  Since  the  velocity  is  -\/~v.>  dt  =  ds  :  *\/~v9  and 
J  ds  <\/  v  =  J  vdt.  The  first  integral  refers  to  momenta  and  the  second 
to  living  forces.  This  duality  enabled  Euler  to  emphasise  that  he  did 
not  offend  the  feelings  of  any  party  to  the  controversy  on  living  forces. 

1  Translated  into  French  by  JOUGUET. 

18 


274  THE   PRINCIPLES  OF  CLASSICAL  MECHANICS 

Euler  verified  that  the  integral  J  ds  <\/~v  =  J  vdt  is  an  extremum  in 
the  parabolic  motion  of  a  particle  subject  to  a  central  force.  He  then 
generalised  this  result  to  a  particle  attracted  by  any  number  of  fixed 
centres. 

Mach  remarks  in  this  connection,  u  Euler,  a  truly  great  man,  lent 
his  reputation  to  the  principle  of  least  action  and  the  glory  of  his  inven 
tion  to  Maupertuis ;  but  he  made  a  new  thing  of  the  principle,  practic 
able  and  useful. "  One  should  observe,  however,  that  Euler  did  not 
condemn  the  doctrine  of  final  causes  as  d'Alembert  had  done.  On  the 
contrary,  the  true  significance  of  an  extremum  principle  should  be,  in 
his  opinion,  sought  in  a  sound  metaphysics.  Indeed,  he  concludes  in 
the  following  terms. 

„  Since  bodies,  because  of  their  inertia,  resist  all  changes  of  state, 
they  will  obey  forces  which  act  on  them  as  little  as  possible  if  they  are  free. 
Therefore,  in  the  motion  generated  the  effect  produced  by  the  forces  will 
be  less  than  if  the  bodies  were  moved  in  some  other  way.  The  strength 
of  this  argument  may  not  be  sufficiently  clear.  If,  however,  it  is  in  accord 
with  the  truth  I  have  no  doubt  that  a  sounder  metaphysics  will  enable 
it  to  be  demonstrated  clearly.  I  leave  this  task  to  others,  who  make  a 
profession  of  metaphysics  (quod  negotium  aliis,  qui  metaphysicam  pro- 
faentur,  relinquo) . " 

12.  FINAL  REMARK. 

To  recapitulate,  Fermat,  in  geometrical  optics,  stated  the  first  mini 
mum  principle  that  was  not  trivial.  He  was  not  able  to  convince  the 
Cartesians  although  he  eventually  accepted  a  reduction  of  his  principle 
to  the  rank  of  a  "  small  geometrical  assistance  "  offered  to  Nature 
without  any  pretension  to  dictate  her  doings. 

No  one  accepted  Format's  conclusion,  however  plausible  it  might 
have  been,  on  the  relative  velocity  of  light  in  dense  and  rare  media. 
Maupertuis  cannot  be  reproached  for  having  shared  the  errors  of  his 
time,  reinforced  as  they  were  by  the  double  authority  of  Descartes  and 
Newton. 

By  means  of  a  very  simple  differential  argument,  Maupertuis  succeed 
ed  in  making  both  the  newtonian  law  of  propagation  and  that  of  refrac 
tion  amenable  to  an  extremum  law. 

Was  the  development  of  his  thought  as  was  said  at  the  time,  of  an 
exclusively  metaphysical  kind  ?  Yes,  if  the  explanation  of  his  motives 
with  which  he  prefaced  his  analysis  is  considered  on  its  own.  I  am 
reluctant  to  suggest  a  more  natural,  but  much  more  worldly,  explana 
tion—that  Maupertuis  had,  in  his  presentation,  reversed  the  order  of 


THE   PRINCIPLE    OF    LEAST   ACTION  275 

the  arguments  ;  that  he  first  discovered  the  differential  argument  which 
we  have  reproduced  and  then  presented  it,  a  posteriori,  as  the  conse 
quence  of  an  economic  principle  which  indicated  both  the  power  and 
the  wisdom  of  the  creator. 

If  this  had  been  the  whole  of  Maupertuis'  contribution,  his  name 
would  have  fallen  into  oblivion,  at  least  as  far  as  the  invention  of  prin 
ciple  is  concerned.  For  in  optics  only  Fermat's  principle,  which  Mau 
pertuis  had  set  out  to  demolish,  has  survived. 

Maupertuis'  extension  of  the  principle  of  least  action  to  dynamics 
appears  rather  gratuitous,  for  it  rests  on  a  fragile  analogy — yet  it  is 
this  principle  which  has  survived  and  assured  the  fame  of  its  author. 
Certainly,  as  early  as  1744,  Eider  gave  the  exact  mathematical  justifi 
cation  of  the  principle  in  the  special  but  important  case  of  the  mechanics 
of  a  particle.  Following  Euler's  example,  Lagrange  stated  the  prin 
ciple  of  the  greatest  or  the  least  living  force  without  Maupertuis.  But 
Euler  himself  was  determined  to  leave  the  honour  of  having  to  disco 
vered  the  principle  of  least  action  to  Maupertuis  ;  and  on  this  fact,  he 
knew  the  evidence. 

The  term  u  least  "  is  only  justifiable  on  the  metaphysical  plane, 
where  every  maximum  would  be  evidence  of  the  imperfection  of  the 
Creator's  wisdom.  Despite  the  criticism  of  Lagrange  and,  later,  that 
of  Hamilton,  the  name  has  survived  and  even  now  it  is  encountered  in 
all  the  books. 

In  the  domain  of  the  laws  of  impact  Maupertuis'  contribution  was 
most  constructive.  His  principle  enabled  him  to  encompass  the  cases 
of  elastic  bodies  and  hard  bodies  which  had  previously  appeared 
separate,  if  not  contradictory.  A  trace  of  this  disjunction  was  still 
apparent  in  Lagrange's  work. 


CHAPTER   SIX 


EULER  AND  THE  MECHANICS  OF  SOLID  BODIES 

(1760) 


In  1760  Euler  published  a  Theoria  motus  corporum  solidorum  seu 
rigidorum.  This  was  eventually  amended  and  added  to  by  his  son,  in 
a  new  edition  which  appeared  in  1790. 

The  treatise  starts  with  an  introduction  in  which  Euler  confirms 
the  principles  of  his  Mechanica  (1736)  .x 

In  connection  with  the  mechanics  of  solids,  Euler  states  that  he 
will  consider  the  characteristic  property  of  a  solid  to  be  the  conservation 
of  the  mutual  distances  of  its  elements.  For  every  solid  he  defines  a  cen 
trum  massae  or  centrum  inertiae,  remarking  that  the  term  "  centre  of  grav 
ity  "  implies  the  more  restricted  concept  of  a  solid  that  is  only  heavy, 
while  the  centre  of  mass  of  inertia  is  defined  by  means  of  the  inertia  alone 
(per  solam  inertiam  determinari) ,  the  forces  to  which  the  solid  is  subject 
being  neglected.  This  apt  comment  has  not  prevailed  against  usage. 

Euler  also  defines  the  moments  of  inertia — a  concept  which  Huyghens 
lacked  and  which  considerably  simplifies  the  language — and  calculates 
these  moments  for  homogeneous  bodies. 

He  systematically  studies  the  motion  of  a  solid  body  about  a  fixed 
axis,  the  given  forces  being  at  first  zero  and  then  being  equated  to  the 
gravity  alone.  He  demonstrates  the  existence  of  spontaneous  or 
permanent  axes  of  rotation  for  a  solid  body  and  thus  clarifies  the  notion 
of  the  principle  axes  of  inertia. 

He  then  investigates  the  motion  of  a  free  solid  by  decomposing 
it  into  the  motion  of  the  centre  of  inertia,  and  the  motion  of  the  solid 
about  the  centre  of  inertia.  Euler  clearly  distinguishes — 

1)  the  variation  of  the  velocity  of  the  centre  of  inertia  I ; 

2)  the  variation  of  the  direction  of  the  point  I ; 

3)  the  variation  of  the  rotation  of  the  solid  about  an  axis  passing 
through  I. 

4)  the  variation  of  this  axis  of  rotation. 
1  See  above,  p.  239. 


THE    MECHANICS    OF   SOLID   BODIES  277 

We  shall  make  this  clear  by  an  analysis  of  problems  which  Euler 
himself  treated. 

"  Problem  86.  —  Being  given  a  solid  body  actuated  by  a  given  angular 
velocity  about  some  axis  passing  through  its  centre  of  inertia,  to  find 
the  elementary  forces  which  must  act  on  the  elements  of  the  solid  in  order 
that  the  axis  of  rotation  and  the  angular  velocity  should  undergo  given 
variations  in  the  time  dt."1 

Let  I  be  the  centre  of  inertia  ;  L4,  IB,  1C  the  principal  [or  central] 
axes  of  the  solid  ;  a,  ft,  y  the  angles  between  the  axis  of  rotation  and 
I  A,  IB,  and  1C;  co  the  angular  velocity  of  rotation  of  the  solid  ;  x,  y,  z 
the  coordinates  of  some  element  of  the  solid  with  respect  to  the  principal 
axes  ;  u,  v9  w  the  components  of  the  velocity  of  this  element  along 
the  same  axes  ;  X,  Y,  Z  the  unknown  force  applied  to  the  particular 
element  considered,  whose  mass  is  dM. 

The  data  of  the  problem  are  da,  d/?,  dy  and  doo  and  the  unknowns, 
X,  Y,  Z.  According  to  the  fundamental  law  of  Euler's  dynamics, 

du,  dv  and  dw  are  proportional  to  -r-r-r,  -rvji  and  -^-.     Therefore  the 
problem  reduces  to  the  calculation  of  du,  dv  and  dw.     Now 

(u  =  co  (z  cos  ft  —  y  cos  y)      (dx  =  udt  =  a>dt  (z  cos  /?  —  y  cos  y 
v  =  co  (x  cos  y  —  z  cos  a)      <  dy  =  ---- 
w  =  CD  (y  cos  a  —  x  cos  /?)      (  dz  =  ---- 

A  simple  differentiation  gives 

du  =  dco  (z  cos  ]8  —  y  cos  y)  —  wzdfi  sin  /?  +  coydy  sin  y 

+  co2dt  (y  cos  a  cos  /3  +  %  cos  a  cos  y  —  x  sin2  a) 
dv  =  ... 
dw  =  .  .  , 

and  the  unknown  forces,  X,  Y,  Z,  applied  to  the  element  dM  (x,y,z) 
are  deduced  from  the  fundamental  law. 

In  the  next  problem  (No.  87)  Eider  calculates  the  moments  P,  (), 
jR,  with  respect  to  the  principal  axes,  of  the  forces  applied  in  the  con 
ditions  specified.  By  the  definition  of  moments, 


dP  =         (ydw  -  zdv)        dQ  =  .  .  .         dR  =  .  .  . 
for  the  element  dM. 

1  Page  337  of  the  new  latin  edition  of  EULER'S  -work,  which  we  follow  here. 


278  THE    PRINCIPLES  OF  CLASSICAL  MECHANICS 

By  summing  over  all  the  solid  body,  after  replacing  du,  dv  and 
dw  by  their  appropriate  value,  Euler  finds 

j 

P  =  —  (Adco  cos  a  —  oAda.  sin  a  -f-  co2  (C  —  jB)  at  cos  P  cos  y). 
/T?\  J  dt 

(EMe=  •  •• 

> 

(A,  JB,  C,  central  moments  of  inertia  of  the  solid.) 

Having  obtained  this  result,  Euler  poses  the   following    problem. 

"  Problem  88.  —  If  a  solid  body,  turning  about  an  axis  passing 
through  its  centre  of  inertia  with  angular  velocity  o>,  is  acted  upon  by  some 
forces,  to  find  the  variation  of  the  axis  of  rotation  and  the  angular  velocity 
at  the  end  of  a  time  dt.  *' 

A  linear  combination  of  the  equations  (E)  gives  the  result  in  the  form 

(C  -  B)  (A  -  C)  (B  -  A) 
da)  = — — oral  cos  a  cos  p  cos  y 

ABC 

IP  cos  a       Q  cos  /5       JR  cos  y 

4-  at  I -\ h  

V     A  B  C 

But  Euler  discovered  that  the  equations  can  be  cast  into  a  much 
more  simple  form  by  introducing  the  components  of  the  angular  velocity 
of  instantaneous  rotation  about  the  central  axes  of  inertia  ;  that  is, 
by  introducing  the  quantities  p  =  a)  cos  a,  q  —  a>  cos  {3  and  r  =  co  cos  y. 

Under  these  conditions  a,  /?  and  y  no  longer  appear  in  the  equations 
(E),  which  take  the  form 


Q  =  B       +  (A  -  C)  rp 


Etder  immediately  saw  the  importance  of  these  equations  —  "  summa 
totius  Theoriae  motus  corporum  rigidorum  his  tribus  formulis  satis 
simplicibus  continebitur.  " 

Thus  Euler's  mathematical  talent  enabled  him  to  discover  the 
equations  which  express  the  general  motion  of  a  solid  body  under 
the  influence  of  arbitrary  forces  by  means  of  the  decomposition  of 
this  motion  into  the  motion  of  the  centre  of  inertia  and  the  rotation 
about  the  centre  of  inertia.  An  essential  part  of  this  process  was 
the  consideration  of  the  central  axes  of  inertia  —  that  is,  the  con 
sideration  of  a  real  moving  reference  frame  fixed  in  the  solid. 


CHAPTER   SEVEN 


CLAIRAUT  AND  THE  FUNDAMENTAL  LAW  OF  HYDROSTATICS 


1.  CLAIRAUT'S  PRINCIPLE  OF  THE  DUCT. 

Clairaut  (1713-1765)  was  led  to  formulate  the  general  law  of  the 
equilibrium  of  a  fluid  mass  by  the  contemporary  investigations  of 
the  figure  of  the  Earth.1 

Clairaut  did  not  openly  take  sides  in  the  conflict  between  the 
doctrine  of  vortices  and  that  of  attraction.  However,  he  remarked 
that  the  Neo-Cartesians,  while  recognising  part  of  the  newtonian  system, 
assumed  a  priori,  whatever  the  shape  of  the  Earth  might  be,  that 
the  gravity  was  inversely  proportional  to  the  square  of  the  distance. 
They  then  compounded  this  gravitational  force  with  the  centrifugal 
force  calculated  for  a  given  shape  of  the  Earth.  In  the  procedure 
of  the  attraction,  on  the  other  hand,  the  law  of  gravity  depended  on 
the  shape  of  the  Earth  itself.  "  The  Newtonians  must  find  a  spheroid 
such  that  a  corpuscle,  placed  at  an  point  on  its  surface  and  which 
is  subject  to  both  the  centrifugal  force  and  the  attractions  of  all  the 
parts  of  the  spheroid,  will  take  a  direction  perpendicular  to  that  sur 
face.  "  2 

Huyghens  assumed  that  the  gravity  must  be  normal  at  each  point 
of  the  surface  of  a  fluid  mass.  Newton  supposed  the  equality  of  the 
weights  of  two  liquid  columns  ending  at  the  centre  of  mass.  Bouguer 
had  the  merit  of  observing,  as  early  as  1734  3,  that  these  two  hypotheses 
were  incompatible  for  certain  laws  of  gravity.  Whence  the  theme 
of  Clairaut's  investigation — 

"  To  find  the  laws  of  hydrostatics  which  agree  equally  with  all 
kinds  of  hypotheses  about  gravity.  "  4 

1  Theorie  de  la  figure  de  la  Terre  tirie,  des  principes  de  Vhydrodynamique  (Durand, 
Paris,  1743). 

2  J/oc.  eft.,  p.  xxj. 

3  Comparaison  des  deux  lois  que  la  Terre  et  les  autres  Planetes  doivent  observer  dans 
la  figure  que  la  Pesanteur  leur  fait  prendre,  Mtimoires  de  FAcad&mie  des  Sciences,  1734. 

4  CLAIRAUT,  Joe.  c£t.,  p.  xxxij. 


280 


THE    PRINCIPLES  OF  CLASSICAL   MECHANICS 


Clairaut  states  the  following  principle  at  the  beginning  of  his  paper. 

"A  mass  of  fluid  cannot  be  in  equilibrium  unless  the  efforts  of  all 
the  parts  which  are  contained  in  a  duct  of  any  shape,  which  is  imagined 
to  traverse  the  whole  mass,  cancel  each  other  out. " l 


0 


N 


He  justifies  this  in  the  following  way. 

"  Since  the  whole  mass  PEpe  is  supposed  to  be  in  equilibrium, 
any  part  of  the  fluid  could  become  solid  without  the  remainder  changing 
its  condition.  Suppose  that  all  the  mass  is  solidified  except  for  what 
is  necessary  to  form  the  duct  ORS.  The  duct  will  therefore  be  in 
equilibrium.  Now  this  can  only  occur  if  the  efforts  of  01?  to  leave  the 
duct  through  S  are  equal  to  those  of  SR  to  leave  through  0.  " 

This  principle  includes  Newton's  hypothesis,  as  may  be  verified  by 
the  consideration  of  a  duct  MCN  passing  through  the  centre  C.  It  also 
includes  Huyghens'  hypothesis —  it  suffices,  indeed,  to  consider  a  duct 
FGD  lying  along  the  surface.  This  duct  may  be  in  equilibrium  in 
two  ways —  the  first,  from  the  fact  of  Huyghens'  hypothesis  itself ; 
the  second,  because  of  the  fact  that  a  part  FG  thrusting  towards  D 
is  compensated  by  a  part  GD  thrusting  towards  F.  But  since  the 
length  of  the  duct  is  arbitrary,  a  small  piece  FG  should  be  in  equilibrium 
just  as  much  as  the  whole  duct,  which  excludes  the  preceding  com 
pensation.  Therefore  it  is  necessary  to  return  to  Huyghens'  hypothesis. 

But  the  most  valuable  form  in  which  Clairaut  stated  his  principle 
is  the  following  one. 

1  CLAIRAUT,  Joe.  tit.,  p.  1. 


CLAIRAUT  AND    HYDROSTATICS 


281 


"  In  order  that  a  mass  of  fluid  may  be  in  equilibrium^  it  is  necessary 
that  the  efforts  of  all  the  parts  of  the  fluid  which  are  contained  in  a  duct 
which  is  re-entrant  upon  itself  should  cancel  each  other  out. "  a 


This  proposition  can  be  justified  immediately  by  solidifying  all 
the  fluid  except  that  in  the  duct  IKLT.  The  efforts  of  the  parts 
IKL,  ITL  must  be  equivalent  to  each  other,  "  or  else  there  would 
be  a  perpetual  current  in  the  duct.  "  It  can  also  be  deduced  from 
the  preceding  proposition  by  the  consideration  of  two  ducts  FIKLG 
andFITLG. 


Fig.  94 


Clairaut  then  observes  that  in  the  consideration  of  two  ducts  ab^ 
a/?,  which  are  filled  with  liquid  and  rotate  about  an  axis  Pp,  the  total 
effort  of  the  centrifugal  force  on  the  duct  ab  will  be  the  same  as  that 

1  CLAIRAUT,  Joe.  cit.,  p.  5. 


282 


THE   PRINCIPLES  OF  CLASSICAL   MECHANICS 


on  the  duct  a/?  if  a  and  a,  b  and  /5  are  respectively  at  the  same  distance 
from  the  axis.  It  follows  that  "  when  it  is  desired  to  investigate 
whether  a  law  of  gravity  is  such  that  a  mass  of  fluid  turning  about 
an  axis  can  preserve  a  constant  shape,  no  purpose  it  served  by  paying 
attention  to  the  centrifugal  force.  That  is,  that  if  the  mass  of  fluid 
can  have  a  constant  shape  when  not  rotating,  it  will  also  be  able  to 
have  one  when  rotating.  " 


If  a  duct  abed  is  considered  in  the  mass,  this  must  be  in  equilibrium 
in  order  that  the  mass  should  have  a  constant  shape.  Now  it  is  seen 
that  the  sum  of  the  effects  of  the  centrifugal  force  on  abed  is  nothing  ; 
for  ab  and  cb  will  thrust  on  b  to  the  same  extent,  just  as  ad  and  cd 
will  thrust  equally  on  d .  Therefore  the  rotation  will  not  prevent  the 
equilibrium  of  a  duct  which  is  re-entrant  upon  itself.  Accordingly, 
"  if  the  duct  is  in  equilibrium  when  only  gravity  alone  is  considered, 
it  will  still  be  in  equilibrium  if  it  is  supposed  that,  instead  of  gravity, 
the  actual  weight,  composed  of  gravity  and  the  centrifugal  force,  is 
considered.  " 


2.  THE  CONDITION  TO  BE  SATISFIED  BY  THE  LAW  OF  GRAVITY  TO  ASSURE 
THE  CONSERVATION  OF  THE  SHAPE  OF  A  ROTATING  FLUID  MASS. 

Clairaut  supposes  that  gravity  has  two  components,  P  and  Q, 
which  are  parallel  to  the  axes  CP  and  CE.  He  considers  an  arbitrary 
duct  ON  which  ends  on  the  surface,  and  an  element  of  this  duct,  Ss9 
which  has  Sr  =  dx  and  sr  =  dy. 


CLAIRAUT   AND   HYDROSTATICS 


283 


The  force  Q  acts  along  SH.     The  projection  of  Q  on  the  direction 


rs 


of  the  duct  has  the  value  Q  •  ^.  By  multiplying  by  the   mass  of  the 

oS 

element,  it  is  found  that  "  the  effort  which  the  force  Q  will  cause 
the  cylinder  to  exert  on  the  point  0"  has  the  value  Qdy.  Similarly, 
the  force  P  gives  the  effort  Pdx.  Whence  the  total  effort  of  gravity  is 


N 


It  is  necessary  that  the  sum  of  the  efforts  of  gravity  on  any  duct 
ON  should  have  the  same  value  as  if  any  other  duct  passing  through 
the  points  0,  N  had  been  taken.  "  The  equilibrium  of  the  fluid  requires 
that  the  weight  of  ON  does  not  depend  on  the  curvature  of  OSN  or, 
that  is,  on  the  value  of  y  as  a  function  of  x.  Therefore  it  is  necessary 
that  Pdx  +  Qdy  can  be  integrated  without  knowing  the  value  of  x  ; 
that  is,  it  is  necessary  that  Pdx  -f-  Qdy  should  be  a  complete  differential, 
or  that 


.  -r=  LJL.   »  1 


dy       dx 


3.  CONDITION  FOR  THE  EQUILIBRIUM  OF  A  FLUID  MASS. 

Clairaut  set  himself  [the  following  problem.2  "  Supposing  that 
the  force  which  actuates  the  particles  of  a  fluid  had  been  decomposed 
into  three  others,  of  which  the  first  acts  perpendicularly  to  the  plane 

1  CLAIRAUT  gave  this  last  condition  in  the  Memoires  de  I* Academic  des  Sciences 
for  1740,  p.  294. 

2  CLAIRAUT,  loc.  cit.,  p.  96. 


284 


THE   PRINCIPLES  OF  CLASSICAL   MECHANICS 


QAP  and  the  second  and  third  along  two  perpendicular  directions, 
QA  and  AP,  in  the  plane  QAP,  it  is  required  to  find  the  relation  there 
must  be  between  these  three  forces  in  order  that  the  equilibrium  of 
the  fluid  may  be  possible.  " 


Fig.  97 

If  P,  Q  and  R  are  the  components  of  the  force  parallel  to  AP, 
AQ  and  AR,  and  Nr9  rs  and  $n  are  represented  by  dx,  dy  and  dz,  then 
an  argument  quite  analogous  to  that  of  the  preceding  §  gives  the 

condition 

Pdx  _j_  Qfy  -|_  Rdz  =  complete  differential 


or 


'By 


Hz 


__ 

3z~~  dy* 


Apart  from  this,  Clairaut  also  defines  the  lines  and  surfaces  of 
intensity  of  gravity.  He  verifies  that  a  fluid  mass  that  is  imagined 
to  be  divided  into  an  infinite  number  of  layers  which  are  defined  by 
the  surfaces  of  intensity,  will  be  in  equilibrium  if,  at  each  point  of  one 
of  the  surfaces,  the  weight  is  inversely  proportional  to  the  thickness 
of  the  layer. 

This  remarkable  analysis,  which  can  be  regarded  as  the  introduction 
of  the  concept  of  potential,  enabled  Clairaut  to  make  an  important 
contribution  to  the  theory  of  the  figure  of  the  Earth.  Newton  had 
assumed,  a  priori,  the  shape  of  an  elliptic  spheroid.  He  considered 
two  columns  —  one  connecting  the  centre  to  a  Pole  and  the  other,  the 
centre  to  the  Equator  —  and  equated  the  difference  in  their  weight  to 
the  sum  of  the  centrifugal  force  on  the  different  portions  of  the  Equitorial 

229 
column.     The    ratio   of  the    axes    obtained    in    this  way  was    TTTTT:- 


The  Neo-Cartesians,  on  their  side,  announced  the  ratio  "r^r. 


CLAIRAUT   AND   HYDROSTATICS  285 

As  early  as  1737  Clairaut  was  able  to  show  that  the  elliptic  spheroid 
was  an  equilibrium  figure. 

230 
Newton's  value  —  or  that  of  MacLaurin  ;rry  —  supposed    the    Earth 


to  be  homogeneous.  This  value  was  disproved  by  experiments  made 
in  Lapland  by  an  expedition  sent  at  the  command  of  the  King.  This 
expedition  consisted  of  four  members  of  the  Academic  des  Sciences  — 
Clairaut,  Camus,  Le  Mourner  and  Maupertuis  —  who  were  joined  by 
two  others  —  Celsius  and  the  Abbe  Outhier.  It  embarked  at  Dunkirk 
on  May  2nd,  1736.1 

The  relative  magnitudes  of  degrees  of  meridian  obtained  in  this  way 

indicated  a  flattening  about  ^-r,   less  than  that  which  Newton  had 


announced.  It  was  therefore  necessary  to  give  up  the  hypothesis 
of  the  homogeneity  of  the  Earth.  If  the  Earth  were  formed  of  similar 
layers,  it  shape  would  not  conform  to  the  fundamental  law  of  hydro 
statics.  And  Clairaut  decided  on  the  existence  of  layers  which  were 
flatter  as  they  were  further  from  the  centre,  the  flattening  following  a 
law  that  depended  on  the  decrease  of  the  density  between  the  centre 
and  the  surface. 

^  l  For  the  account  of  this  mission,  see  MAUPERTUIS'  Discours  lit  dans  VAssembtie 
publique  de  V  Academic  Royale  des  Sciences  sur  la  mesure  de  la  Terre  au  Cercle  polaire 
(CEuvres  completes,  Vol.  Ill,  1756,  p.  89). 


CHAPTER   EIGHT 

DANIEL  BERNOULLI'S  HYDRODYNAMICS 
D'ALEMBERT  AND  THE  RESISTANCE  OF  FLUIDS 

EULER'S  HYDRODYNAMICAL  EQUATIONS 
BORDA  AND  THE  LOSSES  OF  KINETIC  ENERGY  IN  FLUIDS 


1.  RETURN  TO  THE  HYDRODYNAMICS  OF  THE  xviith  CENTURY. 

We  have  already  described  the  attempts  of  Newton  and  Varignon 
to  explain  the  law  which  Torricelli  had  formulated.  We  must  now 
return  to  the  work  of  Mariotte,  who  emerges  as  the  forerunner  of  the 
important  XVIIIth  Century  school  of  hydrodynamics. 

As  early  as  1668  a  Committee  of  the  Academic  des  Sciences  was 
formed  and  instructed  to  verify  TorriceUi's  law  experimentally.  The 
Committee's  members  were  Huyghens,  Picard,  Mariotte  and  Cassini. 
It  extended  its  investigations  to  the  determination  of  the  effect  of 
the  impact  of  a  fluid  stream  on  a  plane  surface. 

Influenced  by  these  investigations,  Mariotte  published,  in  1684, 
the  Traite  du  mouvement  des  eaux  in  which  he  carried  the  subject  further. 
He  verified  Torricelli's  law  without  observing  the  contraction  of  the 
stream.  Newton  corrected  this  error  in  the  second  edition  of  the 
Principia. 

In  the  matter  of  the  impact  of  a  fluid  stream  on  a  surface,  Mariotte 
had  the  merit  of  demonstrating  the  importance  of  the  deviation  from 
the  momentum  of  the  fluid.  But  he  compared  this  problem  with 
the  action  of  a  fluid  current  on  a  completely  immersed  solid,  thus 
disregarding  the  reconstitution  of  the  stream-lines  behind  the  obstacle. 

Mariotte  was  also  the  first  to  introduce  considerations  of  similitude 
in  the  resistance  of  fluids,  and  the  first  after  Huyghens  to  state  that  the 
resistance  of  a  fluid  was  proportional  to  the  square  of  the  velocity. 

Finally,  Mariotte  turned  his  attention  to  hydraulics,  and  studied 
the  velocity  of  flow  in  rivers  or  canals,  and  the  friction  of  water  in 
pipes. 


HYDRODYNAMICS  287 

2.  DANIEL  BERNOULLI'S  HYDRODYNAMICS. 

The  beginning  of  the  XVIIIth  Century  was  a  period  of  extraordinary 
development  for  both  theoretical  and  applied  hydrodynamics.  In  the 
compass  of  the  present  history — limited  to  a  study  of  the  evolution 
of  the  principles  of  dynamics — it  would  be  impossible  to  analyse  the 
complex  development  of  the  mechanics  of  fluids  in  any  detail.  This 
investigation  rapidly  became  an  independent  branch  of  science,  both 
theoretical  and  experimental. 

However,  we  consider  it  valuable  to  deal  with  a  few  typical  achie 
vements  of  the  XVIIIth  Century  in  this  field. 

In  1738  Daniel  Bernoulli  published  Hydrodynamica,  sive  de  viribus 
et  motibus  fluidorum  commentarii.  This  was  a  most  remarkable  work 
which  has  only  aged  a  little  in  the  last  two  centuries. 

In  D,  Bernoulli's  sense,  the  term  hydrodynamica  includes  hydro 
statics — the  science  of  equilibrium — and  hydraulics — the  science  of 
fluid  motion.  "  My  theory  is  novel,  "  he  declared, **  because  it  considers 
both  the  pressure  and  the  motion  of  fluids.  It  might  be  called  hydraulico- 
stattica. " 

D.  Bernoulli's  guiding  principle  was  that  of  the  conservation  of 
living  forces  or,  more  accurately,  that  of  the  equality  between  the  actual 
descent  and  the  potential  ascent  (aequalitas  inter  descensum  actualem 
ascensumque  potentialem) . 

As  the  controversy  on  living  forces  was  then  at  its  height, 
D.  Bernoulli  took  certain  precautions  in  this  respect.  Quite  legiti 
mately,  he  emphasised  that  the  doctrine  of  Leibniz  stemmed,  in  fact, 
from  a  principle  that  Huyghens  had  formulated.  (No  body  freely 
can  rise  to  a  height  greater  than  that  from  which  it  has  fallen.)  * 

Moreover  inelastic  impact,  which  entails  a  loss  of  kinetic  energy, 
has  its  analogy  in  hydrodynamics,  where  it  appears  as  a  reduction 
of  the  ascensus  potentialis.  D.  Bernoulli  excluded  this  occurrence  from 
the  theory,  adding  that  this  was  a  reason  for  applying  the  theory 
with  care. 

Apart  from  the  hypothesis  of  the  conservation  of  living  forces, 
D.  Bernoulli  also  assumed  that  all  the  particles  of  a  slice  of  fluid  which 
was  perpendicular  to  the  direction  of  motion  moved  with  the  same 
velocity,  which  was  inversely  proportional  to  the  cross-section  of  the 
slice.  Further,  he  only  studied  stationary  states  (fingenda  est  unifor- 
mitas  in  motu  aquarum). 

We  shall  give  a  concrete  example  of  one  of  the  numerous  problems 
which  D.  Bernoulli  solved. 

1  See  above,  p.  187. 


288 


THE   PRINCIPLES  OF  CLASSICAL   MECHANICS 


"Let  there  be  a  vessel  of  very  large  cross- section,  ACEB,  which  is 
kept  full  of  fluid,  and  let  it  be  pierced  with  a  horizontal  cylindrical  tube 
ED.  At  the  end  of  the  tube  is  an  orifice  0  through  which  the  fluid  escapes 
with  constant  velocity.  It  is  required  to  find  the  pressure  exerted  on  the 
walls  of  ED. 


B 


Fig.  98 


a  c    r 


I  I 


6    b  d  D 


"  Let  a  be  the  height  of  the  surface  AB  above  the  orifice  o.  When 
the  steady  state  is  established  (siprimafluxus  momenta  excipias)  the  velo 
city  with  which  the  water  leaves  through  o  is  constant  and  equal  to  -y/  a, 
since  we  suppose  that  the  vessel  remains  full.  If  n  is  the  ratio  of  the 

section  of  the  tube  to  that  of  the  orifice,  the  velocity  of  the  water  in 

/ — 

the  tube  will  be  — — .     If  the  whole  end  FD  were  missing  the  velocity 

71 

of  the    water   in   the    tube   would   be  \/a,    which   is    greater   than 
Therefore  the  water  in  the  tube  strives  towards  a  greater  motion, 


71 


to  which  the  end  FD  presents  an  obstacle.  From  this  there  results  an 
over-pressure  which  is  transmitted  to  the  boundary  walls.  The  press 
ure  on  the  boundary  walls  is  thus  proportional  to  the  acceleration 
which  the  fluid  would  take  if  the  obstacle  were  instantaneously  taken 
away  and  the  fluid  were  allowed  to  escape  into  the  atmosphere. 

"  All  this  happens  as  if,  during  the  escape  through  the  orifice 
o,  the  tube  FD  were  suddenly  broken  off  at  cd  and  the  acceleration 
of  the  small  portion  of  fluid  abed  were  sought.  .  .  .  Thus  we  must 
consider  the  vessel  A BEcdC  and,  with  its  help,  try  to  find  the  acceleration 

A/  ^ 
which  the  particle,  of  velocity  — — ,  takes  on  escaping. 


HYDRODYNAMICS  289 

**  Let  v  be  the  variable  velocity  of  the  water  in  the  tube  Ed  ;  n  be 
the  cross-section  of  the  tube  ;  c  its  length,  equal  to  EC  ;  and  let  dx  be 
the  length  ac.  A  portion  of  the  fluid  enters  the  tube  at  E  at  the  same 
time  that  abed  escapes  from  it.  The  portion  at  E,  whose  mass  is  ndx, 
acquires  the  velocity  v  or  the  living  force  nv2dx,  which  is  generated 
suddenly.  Indeed,  since  the  section  of  the  vessel  Ae  is  infinite,  the 
portion  of  the  liquid  at  E  does  not  have  any  velocity  before  entering 
the  tube.  To  the  living  force  nvzdx  will  be  added  the  increase  of  the 
living  force  which  the  water  receives  in  Eb  when  the  portion  ad  escapes  — 
say  2ncvdv.  These  two  quantities  together  are  due  to  the  real  descent 
of  the  portion  of  the  fluid  from  the  height  BE,  or  a.  Therefore 

2ncvdv  =  nadx 


or 

vdv       a  — 


dx  2c 

"  Throughout  the  motion  the  increase  dv  of  the  velocity  is  propor 
tional  to  the  pressure  produced  in  the  time  — .  Therefore  the  press 
ure  which  is  exerted  on  the  portion  ad  is  proportional  to  the  quantity 

dv      ,       .          a  —  t;2 
v  —  ;  that  is,  to 


dx  '  ----     '  2c 

At  the  instant  when  the  tube  is  broken,  v  =  —  —   or  v2  =  —  -. 

n  n2 

71     ~  -       1 


i      — 

This  expression  is  to  be  substituted  in  —  -  -  ,  which  becomes  a 
^  2c 

"  And  it  is  this  quantity  which  is  proportional  to  the  pressure  of  the 
water  on  the  portion  ac  of  the  tube,  whatever  the  section  of  the  tube 
may  be  and  whatever  the  orifice  in  the  end  might  be.  ... 

"  If  the  orifice  is  infinitely  small,  or  n  is  infinitely  large  compared 
with  unity,  it  is  clear  that  the  water  exerts  the  whole  pressure  cor 
responding  to  the  height  a.  This  pressure  we  call  a.  But,  then,  unity 
is  vanishingly  small  compared  with  n2  and  the  quantity  to  which  the 

pressure    is    proportional   has   the   value   —  .  .  .  .     If  the  quantity   —  - 

£c  £>c 

corresponds  to  the  pressure  a,  the  pressure  corresponding  to  the  quantity 

n2  _  i  n2  _  i 

a  -  -  —  will  be  a  -  -  —  ,   which   is   independent   of  c,   Q.E.D.  " 
2n2c  JIT 

There  is  no  need  to  quote  further  from  among  the  demonstrations 
which  D.  Bernoulli  developed  in  a  similar  way.  They  are  remarkable 
for  their  ingenuity  and  all  start  from  the  single  hypothesis  which  we  have 

19 


290  THE    PRINCIPLES  OF  CLASSICAL  MECHANICS 

recorded.  We  notice  that  in  the  same  treatise  D.  Bernoulli  enunciated 
the  theorem  with  which  his  name  is  still  associated,  and  which  appears,  in  a 
different  form,  in  all  the  classical  treatises  on  hydrodynamics.  D.  Bernoulli 
also  treated  the  impact  of  a  fluid  stream  on  a  plane  in  a  way  which 
was  superior  to  that  used  hy  Mariotte.  He  showed  that  this  problem 
was  distinct  from  the  one  which  concerns  the  effect  of  a  fluid  current  on  a 
completely  immersed  solid.  Together  with  the  theoretical  solutions 
of  the  problems  which  he  treated,  D.  Bernoulli  recorded  much  experi 
mental  material  in  support  of  his  demonstrations. 

We  also  remark,  without  being  able  to  devote  a  discussion  to  it, 
the  fact  that  MacLaurin  (in  his  Treatise  on  Fluxions)  and  Jean  Bernoulli 
(in  his  Nouvette  hydraulique)  had  sought  to  dispense  with  the  assumption 
of  the  conservation  of  living  forces  which  D.  Bernoulli  had  made. 

But,  in  the  words  of  Lagrange,  "  M acLaurin's  theory  is  not  very 
rigorous  and  seems  to  be  contrived  in  advance  to  agree  with  the  results 
that  he  wished  to  obtain.  "  As  for  that  of  Jean  Bernoulli,  it  leaves 
much  to  be  desired  in  clarity  and  accuracy  and,  in  d'Alembert's  opinion, 
"  its  general  principle  is  deduced  so  easily  from  that  of  the  conservation 
of  living  forces  that  it  appears  to  be  nothing  else  than  that  same  prin 
ciple  presented  in  another  form ;  and  again,  Jean  Bernoulli  seeks  to 
confirm  his  method  by  means  of  indirect  solutions  supported  by  the 
laws  of  the  conservation  of  living  forces.  " 

3.  D'ALEMBERT  AND  THE  MOTION  OF  FLUIDS. 

D'Alembert's  contribution  to  the  theory  of  fluids  was  both  extensive 
and  important.  In  the  confines  of  this  history  we  must  restrict  our 
selves  to  the  indication  of  the  principles  of  this  work,  and  to  the  citation 
of  some  typical  examples. 

Generally  speaking  d'Alembert  remained  faithful,  in  his  treatment 
of  the  mechanics  of  fluids,  to  the  Discours  Preliminaire  of  his  Traite 
de  Dynamique.  Nevertheless  he  did  not  carry  into  this  field  the  convic 
tion  that  science  could  be  of  a  purely  rational  origin. 

He  wrote,  "  The  mechanics  of  solid  bodies  depends  only  on  meta 
physical  principles  which  are  independent  of  experiment.  Those 
principles  which  must  be  used  as  the  foundation  for  others  can  be 
determined  exactly.  The  foundation  of  the  theory  of  fluids,  on  the 
other  hand,  must  be  experiment,  from  which  we  receive  only  a  very 
little  enlightenment.  " 

After  having  devoted  long  and  laborious  efforts  to  an  attempt 
at  elucidating  the  motion  and  resistance  of  fluids,  d'Alembert  remained 
optimistic  about  the  future  of  this  theory.  "  When  I  speak  of  the 


HYDRODYNAMICS  291 

limitations  by  which  the  theory  must  be  prescribed,  "  he  wrote,  "  I 
only  contemplate  the  theory  with  such  assistance  as  it  can  now  obtain, 
and  not  the  theory  as  it  may  be  in  the  future,  aided  by  what  is  yet 
to  be  discovered.  For,  in  whatever  subject  one  might  be,  one  should 
not  be  too  ready  to  erect  a  wall  of  separation  between  nature  and  the 
human  mind.  "  1 

However,  d'Alembert  did  not  conceal  the  great  difficulties  that  arose 
in  the  translation  of  such  complicated  phenomena  as  those  of  the 
motion  and  the  resistance  of  fluids  into  a  rational  language.  In  this 
period  of  enlightenment,  when  all  men,  including  d'Alembert,  were 
willing  to  trust — perhaps  too  readily — in  the  universal  validity  of 
science,  he  made  the  reservation — "  not  to  exalt  the  algebraic  formulae 
into  physical  truths  or  propositions  too  readily.  .  .  .  Perhaps  the  spirit 
of  calculation  which  has  displaced  the  spirit  of  system  rules,  in  its 
turn,  a  little  too  strongly.  For  in  each  century  there  is  a  dominant 
style  of  philosophy.  This  style  almost  always  entails  some  prejudice, 
and  the  best  philosophy  is  the  one  which  has  least  of  this  consequence. "  2 

In  1744,  one  year  after  the  publication  of  his  Traite  de  Dynamique, 
d'Alembert  wrote  a  Traite  de  FEquilibre  et  du  Mouvement  des  Fluides. 
In  it  he  refused  to  start  from  the  principle  of  the  conservation  of  living 
forces,  as  Daniel  Bernoulli  had  done  in  his  Hydrodynamica.  For  this 
principle,  as  we  know  from  the  discussion  in  the  Discours  Preliminaire, 
d'Alembert  regarded  as  not  being  sufficiently  well-established  to  be 
used  at  the  basis  of  hydrodynamics.  "  One  of  the  greatest  advantages 
that  follow  from  our  theory  is  that  of  being  able  to  show  that  the  well- 
known  law  of  mechanics  called  the  conservation  of  living  forces  is  as 
appropriate  in  the  motion  of  fluids  as  in  that  of  solid  bodies.  " 

D'Alembert,  a  keen  critic  of  his  predecessors,  reproached  Daniel 
Bernoulli  for  not  having  brought  forward,  in  Volume  II  of  the  Memoires 
de  Petersbourg  (1727),  "  other  evidence  for  the  conservation  of  living 
forces  in  the  fluids  than  that  a  fluid  could  be  regarded  as  aggregate 
of  fluid  particles  which  press  on  each  other,  and  that  the  conservation 
of  living  forces  is  generally  accepted  to  be  applicable  to  the  impact 
of  a  system  of  bodies  of  this  kind.  .  ,  .  Therefore  it  seemed  to  me 
that  it  is  necessary  to  prove,  more  clearly  and  exactly,  the  question  of 
whether  the  principle  is  applicable  to  fluids.  I  had  tried  to  demonstrate 
this,  in  a  few  words  at  the  end  of  my  Traite  de  Dynamique.*  Here  will 
be  found  a  more  detailed  and  more  extended  proof.  "  4 

1  Essai  d'une  nouvelle  thforie  de  la  resistance  des  fluides   (David,  Paris,    1752), 
p.  xxxiv. 

2  Ibid.,  p.  xlj. 

3  Traite  de  Dynamique  (1758  ed.),  p.  269. 

4  Trait^  de  VEquilibre  et  du  Mouvement  des  Fluides  (1744),  Preface. 


292  THE    PRINCIPLES  OF  CLASSICAL  MECHANICS 

At  the  beginning  of  the  Traite  des  Fluides  d'AIembert  explicitly 
stated  the  hypothesis  of  flow  by  parallel  slices  of  fluid,  whose  parallelism 
was  conserved  throughout  the  motion.  Moreover,  he  assumed  that 
all  the  points  of  the  same  slice  had  the  same  velocity.  Necessarily 
this  hypothesis  restricted  the  generality  of  his  analysis. 

Given  these  hypotheses,  d'AIembert  extended  to  fluids  the  principle 
that  he  had  used  as  a  basis  for  his  dynamics. 

"  In  general,  let  the  velocities  of  the  different  slices  of  the  fluid, 
at  the  same  instant,  be  represented  by  the  variable  v.  Imagine  that 
dv  is  the  increment  of  the  velocity  in  the  next  instant,  the  quantities  dv 
being  different  for  the  different  slices,  positive  for  some  and  negative 
for  others.  Or,  briefly,  imagine  that  v  +  dv  expresses  the  velocity 
of  each  layer  when  it  takes  the  place  of  that  which  is  immediately 
below.  I  say  that  if  each  layer  is  supposed  to  tend  to  move  with 
an  infinitely  small  velocity  +  dv,  the  fluid  will  remain  in  equilibrium. 

"  For  since  v  =  v  +  dv  +  dv,  and  the  velocity  of  each  slice  is 
supposed  not  to  change  in  direction,  each  layer  can  be  regarded,  at 
the  instant  that  v  changes  to  v  +.  dv,  as  if  it  had  both  the  velocity 
v  +  dv  and  the  velocity  +  dv.  Now,  since  it  only  retains  the  first 
of  these  velocities,  it  follows  that  the  velocity  +  dv  must  be  such 
that  it  does  not  affect  the  first  and  is  reduced  to  nothing.  Therefore 
if  each  slice  were  actuated  by  the  velocity  +  dv  the  fluid  would  remain 
at  rest.  "  1 

D'AIembert  accompanied  this  theorem  with  the  following 
observation. 

"  If  it  is  supposed  that  the  particles  of  the  fluid  are  subject  to  an 
accelerating  force  9?,  different  for  each  slice  if  so  desired,  then  it  is 
clear  that  at  the  end  of  the  instant  dt  the  velocity  v  will  be  v  +  cpdt 
if  the  slices  do  not  interact  in  any  way.  Therefore,  if  at  the  end  of 
the  instant  dt  the  velocity  v  becomes  v  If  dv  because  of  the  interaction 
of  the  slices,  it  would  be  necessary  to  suppose  that 

v  +  cpdt  =  v  +  d v  +  (pdt  +  dv 

and  it  is  clear  that  the  fluid  would  remain  in  equilibrium  if  only  the 
slice  were  actuated  with  the  velocity  <pdt  +  dv.  "  2 

Starting  from  this  principle,  d'AIembert  went  back  to  the  problems 
which  had  been  studied  by  Daniel  Bernoulli  and  treated  them  anew. 
We  shall  pause  on  a  single  example,  the  first  and  the  most  simple 
which  d'AIembert  treated. 

The  question  is  that  of  the  flow  in  parallel  slices  of  a  fluid  which 

1  Traite  de  rEquilibre  et  du  Mouvement  des  Fluides  (1744),  p.70. 

2  Ibid.,  p.  71. 


HYDRODYNAMICS 


293 


is  supposed  incompressible  in  a  vessel  of  any  shape.  The  licpiid  is 
homogeneous,  has  no  weight,  and  is  set  in  motion  by  an  unspecified 
means ;  for  example,  by  the  impulsion  of  a  piston. 

Let  u  be  the  velocity  of  the  layer  GH.     That  of  the  layer  CD  will 
,     uGH 
beW 

If  y  is  the  width,  and  x  the  side,  of  one  of  the  slices,  then  it  is  possible 
to  write  ydx  =  constant. 


B 


b 
Fig.  99 


Between  the  instant  t  and  the  instant  t  +  dt  the  fluid  moves  from 
the  position  CDLP  to  the  position  cdlp.  If  v  is  the  variable  which 
represents  the  velocity  of  a  slice  in  the  position  CDLP  and  if  v  —  dv  is 
(algebraically)  the  value  of  the  same  variable  at  cdlp?  the  fluid  will 
remain  in  equilibrium  if  each  slice  tends  to  move  with  the  velocity  dv. 
D'Alembert  expresses  this  by 

J  dvdx  =  0 


294  THE   PRINCIPLES   OP  CLASSICAL   MECHANICS 

or  f  ydx  •  dv      n 


uGH 
But  v  = .     Therefore 

y 


f  ydx*vdv  _ 

J  ~H~  =  °* 


Now  "  GH  is  constant  as  well  as  «,  with  respect  to  the  variables 
v  and  dv  "  and  ydx  is  constant.  It  follows  therefore  that 

fyfa.V*  =  fydx-v*. 

Here  V  is  the  velocity  of  each  slice  in  the  instant  which  follows  that 
at  which  its  velocity  is  v. 

"  Therefore  it  is  seen  that  the  principle  of  conservation  of  living  forces 
also  applies  to  fluids.  " 

To  (TAlembert  however,  this  principle  was  a  corollary  which  had 
to  be  verified  in  each  instance.  And  indeed,  he  was  not  surprised 
to  retrieve,  by  means  of  his  principle,  the  solutions  of  the  problems 
which  had  been  treated  by  D.  Bernoulli.  In  this  connection,  d'Alembert 
makes  the  following  observation  in  the  preface  to  his  Traite  des  Fluides. 

"  And  indeed  I  am  forced  to  confess  that  the  results  of  my  solutions 
always  agree  with  those  of  M.  Daniel  Bernoulli.  Nevertheless  it  is 
necessary  to  except  a  small  number  of  these  problems.  These  are 
problems  in  which  that  skilful  geometer  used  the  principle  of  the  conser 
vation  of  living  forces  to  determine  the  motion  of  a  fluid  in  which 
there  is  a  portion  whose  velocity  increases  suddenly  by  a  finite  quantity. 
Such,  for  example,  is  the  problem  in  which  the  question  is  to  find  the 
velocity  of  a  fluid  leaving  a  vessel  which  is  kept  filled  to  the  same 
height.  It  is  supposed  that  the  small  sheet  of  fluid  which  is  added 
at  each  instant  receives  its  motion  from  the  fluid  below,  by  which  it 
is  drawn  along.  It  is  clear  that  in  a  similar  hypothesis  this  sheet 
of  fluid,  which  had  no  velocity  at  the  instant  that  is  was  added  to 
the  surface,  in  the  next  instant  receives  a  finite  velocity  equal  to  that 
of  the  surface  which  draws  it  along.  Now,  without  wishing  to  ask 
whether  or  not  this  hypothesis  is  in  conformity  with  nature,  it  is  certain 
that  the  principle  of  living  forces  should  not  be  used  to  investigate 
the  motion  of  any  system  when  it  is  supposed  that  there  is  some  body 
in  this  system  whose  velocity  varies  in  an  instant  by  a  finite  quantity.  " 

The  Traite  des  Fluides  also  contains  an  attempt  to  investigate 
the  resistance  of  fluids.  D'Alembert  used  rather  a  flimsy  mechanical 
model,  analogous  to  that  which  he  had  reproached  Daniel  Bernoulli 
for  using,  in  order  to  substantiate  the  conservation  of  living  forces. 


HYDRODYNAMICS  295 

"  First  I  determined  the  motion  that  a  solid  body  must  communicate 
to  an  infinity  of  small  balls  with  which  it  was  supposed  to  be  covered. 
Then  I  showed  that  the  motion  lost  by  this  body  in  a  given  time  was 
the  same  whether  it  collided  with  a  certain  number  of  balls  at  once, 
or  whether  it  merely  collided  with  them  in  succession.  Further, 
I  showed  that  the  resistance  would  be  the  same  when  the  corpuscles 
had  some  other  shape  than  the  spherical  one,  and  when  they  were 
arranged  in  any  manner  that  might  be  desired,  provided  that  the  total 
mass  of  these  small  bodies  which  was  contained  in  a  given  space  remained 
the  same.  By  this  means  I  arrived  at  the  general  formulae  for  the 
resistance,  in  which  there  only  appeared  the  relationships  of  the  den 
sities  of  the  fluid  and  the  body  moving  in  it.  "  x 

Finally,  d'Alembert  devoted  a  chapter  to  "  fluids  that  move  in 
vortices  and  to  the  motion  of  bodies  which  are  immersed  in  them.  " 
He  did  not  make  this  study  to  bolster  up  "  a  cause  as  desperate  as  that 
of  Descartes'  vortices,  "  but  because  the  subject  seemed  in  itself  to 
be  "  rather  curious,  independently  of  any  application  to  the  case  of 
planets  that  one  might  desire  to  make.  " 

4.  D'ALEMBERT  AND   THE   RESISTANCE   or   FLUIDS  —  His   PARADOX. 

The  Essai  d'une  nouvelle  theorie  de  la  resistance  des  fluides  (Paris, 
David,  1752)  has  its  origin  in  d'Alembert's  participation  in  the  competi 
tion  held  by  the  Academy  of  Berlin  in  1750.  The  subject  chosen 
concerned  the  theory  of  the  resistance  of  fluids.  The  prize  was  not 
awarded,  the  Academy  having  required  the  authors  to  give  proof 
of  the  agreement  of  their  calculations  with  experiment  by  means  of 
a  supplement  to  their  work. 

D'Alembert,  who  had  competed,  seems  to  have  become  rather 
bitter  at  this  decision,  for  the  observations  available  at  that  time 
were  often  contradictory  and  the  theory  was  sufficiently  difficult 
to  require  a  worker's  undivided  attention.  His  Essai  is,  apart  from 
some  details,  his  contribution  to  the  Berlin  competition. 

In  his  Essai  d'Alembert  obtains,  at  least  for  plane  motions,  the 
general  equations  of  the  motion  of  fluids.2  However,  his  analysis 
is  so  long  and  tortuous  that  we  cannot  attempt  to  summarise  it.  In 
this  achievement  d'Alembert  preceded  Euler  by  some  years — but 
he  did  not  succeed  in  presenting  the  equations  of  hydrodynamics  in 
the  "  direct  and  luminous  "  3  way  that  Euler  was  able  to  discover. 

1  Traite  de  V£quilibre  et  du  Mouvement  des  Fluides,  Preface. 

2  See  also  Opuscules  mathematiques  by  d'ALEMBERT,  especially  Vol.  VI,  p.  379 
(1778  ed.). 

8  LAGRANGE,  Mecanique  analytique.  Part  II,  Section  X. 


296 


THE   PRINCIPLES   OF  CLASSICAL  MECHANICS 


Faithful  to  his  principles,  cPAlembert  reduced  the  search  for  the  resis 
tance  to  the  laws  of  equilibrium  between  the  fluid  and  the  body.  The 
resistance  was  given  by  the  momentum  lost  by  the  fluid. 

Rather  than  attempt  to  analyse  the  difficult  Essai  sur  la  resistance 
des  fluides?  we  consider  it  more  valuable  to  reproduce  a  portion  of  the 
work  in  which  d'Alembert,  guided,  it  seems,  by  his  logical  insight 
alone,  discovers  the  hydrodynamical  paradox  with  which  his  name 
is  still  associated. 

"  Paradoxe  propose   aux  geometres   sur   la   resistance    des   fluides.1 

66  Suppose  that  a  body  is  composed  of  four  equal  and  similar  parts 
and  placed  in  an  indefinite  fluid  contained  in  a  rectilinear  vessel. 

"  Imagine  that  the  body  is  fixed  and  immoveable,  and  that  the 
parts  of  the  fluid  all  receive  an  equal  impulsion  parallel  to  the  sides 
of  the  vessel  and  the  axis  of  the  body.  It  is  clear  that  the  parts  of 
the  fluid  at  the  front  of  the  body  must  be  deflected  and  must  slide 
along  the  body,  thus  describing  curves  which  are  more  like  straight 
lines  as  they  are  further  from  the  body ;  this  up  to  a  certain  distance, 
which  will  be  at  least  that  of  the  boundary  walls  of  the  vessel,  where 
the  parts  of  the  fluid  will  move  in  straight  lines. 

"  If  the  solid  is  not  terminated  in  a  very  sharp  point,  in  such  a 

dy 

way  that  the  derivative   -f-   may  be  either  finite  or   infinite    at   the 

dx 

origin,  there  will  be  or  there  may  be  a  small  portion  of  the  liquid  which 
is  stagnant  at  the  front  of  the  body.2  But  in  order  to  avoid  this  diffi- 

1  Opuscules  mathematiques.  Vol.  V,  p.  132. 

2  We  reproduce  here  the  diagram  used  by  d'ALEM- 
BERT.     The  "  stagnant  part,  "  which,  in  d'ALEMBERx's 
opinion,  -will  exist  in  general,  is  referred  to  in  the  Essai 
sur  la  resistance  des  fluides,  in  the  following  terms — 

"  All  moving  bodies  which  change  direction  only  do 
so  by  imperceptible  degrees.  The  particles  which  move 
along  TF  do  not  travel  as  far  as  A  because  of  the  right 
angle  TA& —  they  leave  TF  at  F,  for  example.  There 
fore,  in  front  of  and  behind  the  solid,  there  are  spaces 
in  which  the  fluid  is  necessarily  stagnant.  " 


Fig.  100 


HYDRODYNAMICS  297 

c?v 

culty,  I  suppose  that  -j-  =  0  at  the  origin,  so  that  the  point  of  the  body 
u/x 

is  infinitely  sharp.  Then  there  will  no  longer  be  stagnant  fluid,  and 
the  fluid  will  run  along  the  forward  surface  as  far  as  the  point  at  which 
dy  again  becomes  parallel  to  the  axis. 

44  With  regard  to  the  backward  part,  it  seems  at  first  sight  that 
the  motion  must  be  different  from  that  at  the  forward  part.  For  at 
the  forward  part  the  fluid  is  not  free  to  follow  its  original  direction, 
while  it  is  so  free  at  the  backward  part.  However,  for  the  moment 
suppose  that  the  particles  of  the  fluid  do  have  the  same  motion  at 
this  backward  part  as  at  the  front.  It  is  easily  discovered,  by  our  theory 
of  the  motion  of  fluids,  that  under  this  supposition  the  laws  of  the 
equilibrium  and  the  incompressibility  of  the  fluid  will  be  fully  satisfied. 
For  since  the  back  is  similar  to  the  front  (hyp.),  it  is  easy  to  see  that 
the  same  values  of  p  and  q  I  that  will  yield  the  equilibrium  and  the 
incompressibility  of  the  forward  part  will  yield  the  same  results  for 
the  backward  part.  Therefore  there  is  nothing  but  justification  for 
the  supposition  concerned.  Therefore  the  fluid  can  move  in  this 
way  at  the  back.  Now  if  it  can,  it  must.  For  there  is  only  one  possible 
way  in  which  the  fluid  can  be  moved  by  the  body. 

"  In  this  condition,  the  fluid  will  exert  a  pressure  on  the  body, 
though  this  pressure  will  not  have  the  effect  of  separating  the  body 
from  its  position  because  the  body  is  immoveable  and  fixed  at  rest 
in  the  middle  of  the  fluid  (hyp.).  Let  u  be  the  velocity  imparted 
to  the  fluid.  The  pressure,  if  it  exists  at  the  first  instant,  will  be 
exerted  on  the  front  of  the  body  and  will  be  &u,  where  k  is  a  quantity 
which  depends  on  the  shape  of  the  body. 

44  Let  uq  be  the  velocity  of  the  fluid  parallel  to  the  axis  and  let  up 
be  the  velocity  of  the  fluid  perpendicular  to  the  axis.  Since  it  is 
supposed  that,  at  the  first  instant,  with  velocity  u  parallel  to  the  axis 
and  that  it  changes  this  velocity  into  the  velocities  uq  and  wp,  the 
fluid  will  exert  a  pressure  on  the  body  which  will  be  the  same  as  if 
the  fluid  were  at  rest  and  the  body  were  moved  with  velocities  u — uq 
and  u—up.  Now  because  of  the  velocity  w,  the  pressure  exerted  will 
be,  by  the  principles  of  hydrostatics,  equal  to  Mu  and  in  the  opposite 
direction  to  u,  if  M  is  the  mass  of  the  body.  And  because  of  the 
velocities  — uq  and  — up,  the  pressure  exerted  will  be  in  a  direction 
opposite  to  that  of  Mu  and  will  be  equal  to  4u  /  dy  J  ds  \/(p*  +  g2)- 
It  is  easy  to  see  this  by  supposing  that  /  ds  *\/(p2  +  q2)  —  0  and 
J  dy  J  ds  \/(p*  +  q2)  =  0  at  the  point  at  which  dy  =  0  and  which 

1  Quantities  proportional  to  the  components  of  the  velocity  in  d'ALBMBERT's 
theory. 


298  THE   PRINCIPLES  OF  CLASSICAL   MECHANICS 

is  not  the  summit  of  the  body — that  is,  at  the  point  at  which  the  tangent 
is  parallel  to  the  axis.  For  it  is  clear  that,  since  dy  =  0  (hyp.)  for 
x  =  0  and  the  body  is  composed  of  four  equal  and  similar  parts,  there 
must  be  a  point  on  each  side  of  the  axis,  and  in  the  centre  of  the  axis, 
where  dy  =  0.  Therefore,  if  it  is  supposed  that  J"  dy  J  ds  \/(p2  -f-  g2) 
=  J?,  in  its  totality,  then  k  =  4R  —  M. 

"  In  the  following  instants  parts  of  the  fluid  evidently  retain  the 
velocities  uq  and  up,  from  which  it  is  easy  to  see,  by  means  of  our 
theory  on  the  resistance,  that  the  pressure  of  the  fluid  on  the  body 
will  be  absolutely  nothing.  For  the  pressure  on  the  forward  surface 
is  equal  and  opposite  to  the  pressure  on  the  backward  surface. 

*4  And  if  it  were  supposed  that  a  force  y,  constant  or  variable  from 
one  instant  to  another  but  always  the  same  for  all  parts  of  the  fluid 
at  the  same  instant,  acted  on  all  these  parts,  they  would  nevertheless 
continue  to  describe  the  same  lines  with  a  velocity  that  would  be 
increased  in  the  ratio  of  ydz  to  u.  And  from  this  it  would  only  result 
that  a  new  pressure  equal  to  ky  was  exerted  on  the  surface  of  the  body. 

"  Suppose  now  that  the  fluid  is  at  rest  and  the  body  is  moved 
along  in  it  with  the  velocity  v,  of  which  it  only  retains  a  part  u.  Give 
the  whole  system — fluid  and  body — a  motion  u  in  the  opposite  direction. 
The  body  will  be  at  rest  in  fixed  space.  In  front  there  will  be  the  force 
M  (v  —  u),  while  the  fluid  will  exert  a  pressure  ku  on  the  body.  The 
latter  will  nullify  the  former. 

"  Therefore 

7          /           \  n/r                       Mv          Mv 
ku  =  (v  —  u)  M  u  — = 

k  +  M       4B 

"  Now  suppose  that  the  body  continues  to  move  in  the  fluid  with 
a  velocity  which  decreases  by  an  amount  ydt  at  each  instant  so  that 
du  =  —  ydt.  Also  suppose  that  the  system  of  body  and  fluid  moves 
on  the  opposite  direction  with  this  decreasing  velocity.  It  is  apparent 
that  the  body  will  be  at  rest  and  that  the  pressure  at  each  instant 
will  be  ky,  or  kdu,  which  must  counterbalance  Mdu.  Therefore 

Mdu  =  du  (4H  —  M)     or     Mdu  ==  2Rdu. 

"  Therefore   either  du  ==  0   or  2R  =  M. 

"  If  du  =  0,  the  body  will  move   uniformly  and  it   will  be  true 

Mv 
that  u  =  — . 

"  If  2R  =  M,  then  u  =  -  and  the  quantity  du  remains  indeter 
minate.  It  could  be  supposed  to  be  zero,  and  even  must  be  supposed  to 


HYDRODYNAMICS  299 

be  zero,  since  there  is  no  other  equation  to  determine  it  than  Mdu  =  2Rdu. 

"  Thus  the  greatest  alteration  that  could  occur  in  the  original  mo 

tion  of  the  body  is  that  the  velocity  v  —  which  is  supposed  to  have  been 

impressed  on  it  —  should  be  changed  to  -—  in  the  first  instant,  and  after 
this  the  body  will  move  without  suffering  any  resistance  due  to  the  fluid. 


_ 
"  If  the  shape  of  the  body  is  such  that  J  dy  J  ds  \/(p*  +  q*)  =  R=  — 

then  u  =  v.  Whence  Mdu  =  du  (4J?  —  M)  =  0  and  therefore  the  body 
will  not  lose  any  velocity  in  the  first  instant.  This  seems  also  borne 
out  by  experiment. 

46  I  do  not  ask  whether  the  quantities  p  and  q  which  are  obtained 
by  the  theory  are  such  that  4>R  =  M  for  any  shape  of  the  body  —  it 
appears  rather  doubtful  that  this  should  be  so.  Neither  do  I  ask 
whether  4J?  could  be  greater  than  M  for  some  shapes  and  less  than  M 
for  others.  These  conditions  would  imply  u  <  v  (contrary  to  exper 
iment)  and  u  >  v  (contrary  to  common  sense).  Therefore  we  will 
have  du  =  0  and,  from  our  theory,  it  will  follow  that  the  body,  supposed 
to  be  of  four  equal  and  similar  parts,  will  suffer  no  resistance  from 
the  fluid. 

"  And  whatever  relation  there  is  supposed  to  be  between  4.R  and  M, 
it  is  apparent  that  the  velocity  v  will,  at  the  most,  only  experience 
an  alteration  in  the  first  instant,  and  that  it  will  then  remain  uniform. 
This  would  be  much  worse  if  4sR  <  M  for  then  the  initial  velocity 
would  first  increase  and  afterwards  remain  uniform. 

"  I  must  therefore  confess  that  I  do  not  know  how  the  resistance  of 
fluids  can  be  explained  by  the  theory  in  a  satisfactory  way.  On 
the  contrary,  it  seems  to  me  that  this  theory,  handled  with  all  possible 
rigour,  yields  a  resistance  which  is  absolutely  nothing  in  at  least  several 
situations.  I  bequeath  this  strange  paradox  to  the  geometers,  that 
they  may  explain  it.  " 


5.    EULER  AND  THE  EQUILIBRIUM  OF  FLUIDS. 

In  a  paper  given  to  the  Academy  of  Berlin  in  1755,1  Euler  directed 
his  attention  to  the  equilibrium  of  fluids. 

He  considered  a  fluid,  either  compressible  or  not,  which  was  subjected 
to  any  given  forces.  "  The  generality  that  I  include ,"  he  declared, 
"  instead  of  dazzling  us,  will  rather  discover  the  true  laws  of  Nature 

1  Principes  g£neraux  de  Fetat  tfequilibre  des  fluides,  Memoires  de  V Academic  de 
Berlin,  1755,  p.  217. 


300  THE   PRINCIPLES   OF  CLASSICAL   MECHANICS 

in  all  their  splendour,  and  there  will  be  found  yet  stronger  reasons 
for  wondering  at  their  beauty  and  simplicity.  " 

The  general  problem  which  Euler  poses  is  the  following  one. 

"  The  forces  which  act  on  all  the  elements  of  the  fluid  being  given, 
together  with  the  relation  which  exists  at  each  point  between  the  density 
and  the  elasticity  of  the  fluid,  to  find  the  pressures  that  there  must  be,  at  all 
points  of  the  fluid  mass,  in  order  that  it  may  remain  in  equilibrium.  " 

In  the  fluid  mass  Euler  considers  an  elementary  rectangular  parallel- 
ipiped  with  one  corner  at  the  point  Z,  of  coordinates  x,  y,  z  and  with 
sides  dx,  dy  and  dz. 

The  components  of  the  "  accelerative  force  "  applied  to  each  element 
are  called  P,  Q  and  R,  and  q  is  the  density  of  fluid  at  Z. 

Then  the  element  of  volume  dxdydz  is  subject  to  the  "  motive 
force  "  whose  components  are 

Pqdxdydz        Qqdxdydz         Rqdxdydz. 
If  p  is  the  unknown  pressure  at  the  point  Z,  then 

dp  =  Ldx  +  Mdy  +  Ndz. 

By  a  very  simple  geometrical  argument  —  which  has  become  class 
ical  —  Euler  deduces  the  general  conditions  of  equilibrium 

L  =  Pq         M  =  Qp         N  =  Rq. 

If  L,  M  and  N  are  the  partial  derivatives  of  a  function  p  (x,  y,  z), 
equilibrium  requires  the  conditions 


d  (Pq)  =  d  (Qq)     d_(Qq^^d^q)     d  (Rq)  =  d  (Pq) 

dy  dx  dz  dy  dx  dz 

If  p  is  a  given  function  of  q  at  each  point  of  the  fluid,  the  relation 
dp  =  q  (Pdx  +  Q  dy  +  Rdz) 

shows  that  Pdx  +  Qdy  +  Rdz  is   the  total  differential  of  the  func- 

.       dp 
tion  —  . 

q 

This  differential  represents  the  "  effort  "  or  the  "  efficacy  "  of 
the  given  force  —  this  was  a  notion  which  Euler  used  in  the  case  of 
central  forces. 

In  a  second  paper,  which  will  be  discussed  in  the  next  §,  Euler 
deduced  a  general  conclusion  from  the  equation  of  equilibrium 

^  =  Pdx  +  Qdy  +  Rdz. 


HYDRODYNAMICS  301 

**  The  forces  P,  (),  R  must  be  such  that  the  differential  form  Pdx  -f 
Qdy  +  Rdz  either  becomes  integrable  when  the  density  q  is  constant 
or  uniquely  dependent  on  the  elasticity  p,  or  becomes  integrable  when 
multiplied  by  some  function.  " 

Euler  did  not  refer  to  Clairaut  in  this  connection.  Although  he 
had  the  merit  of  introducing  the  pressure  and  relating  it  to  the  acceler- 
ative  force  at  each  point  it  must  be  observed,  with  Lagrange,  that 
Euler 's  achievement  was  that  of  applying,  by  generalising  it,  the 
principle  of  Clairaut. 


6.  EULER  AND  THE  GENERAL  EQUATIONS  OF  HYDRODYNAMICS. 

We  now  come  to  a  fundamental  paper  of  Euler  on  the  equations 
of  hydrodynamics.1  So  perfect  is  this  paper  that  not  a  line  has  aged. 

In  assuming  this  difficult  task,  Euler  declared,  "  I  hope  to  emerge 
successful  at  the  end,  so  that  if  difficulties  remain  they  will  not  be 
in  the  field  of  mechanics,  but  entirely  in  the  field  of  analysis.  " 

Euler  considers  a  fluid  which  is  compressible  or  incompressible, 
homogeneous  or  inhomogeneous.  Its  original  state  —  that  is,  the  arrange 
ment  of  the  particles  and  their  velocities  —  is  supposed  known  at  a  given 
instant,  as  are  the  external  forces  acting  on  the  fluid. 

It  is  necessary  to  determine,  at  all  times,  the  pressure  at  each 
point  of  the  fluid  together  with  the  density  and  the  velocity  of  the 
element  passing  through  that  point. 

In  order  to  study  the  present  state  of  the  fluid,  Euler  uses  the  com 
ponents  of  the  accelerative  force,  P,  Q,  R  which  are  known  functions 
of  #,  y,  z  and  Z.2 

The  density  q,  the  pressure  p,  and  the  components  u,  v9  w  of  the 
velocity  of  the  element  of  the  fluid  which  is  at  the  point  Z  at  the  time  * 
are  unknown. 

During  the  time  dt  the  element  of  fluid  at  Z  will  be  carried  to  the 
point  Z',  whose  coordinates  will  be 


x  +  udt        y  +  vdt         z 
The  element  of  fluid  at  2,  of  coordinates 

x  +  dx        y  +  dy         z  +  dz 

1  Principes  g$n$raux  du  mouvement  des  fluides,  Memoires  de  VAcademie  de  Berlin, 
1755,  p.  274. 

2  EULER  also  refers  to  a  variable  r,  the  "  heat  at  the  point  Z,  or  that  other  property 
which,  apart  from  the  density,  affects  the  elasticity.  " 


302  THE   PRINCIPLES  OF  CLASSICAL   MECHANICS 

has  a  velocity  whose  components   are 

.   du  ,         du  _         du  , 
u  +  —  dx  +  —  dy  +  -5-  dz 

dx  dy  dz 

v+   ... 
w  + 

and,  during  the  time  df,  is  carried  to  the  point  z'.  In  order  to  perform 
the  calculation,  Eider  first  considers  a  segment  Zz  which  is  parallel 
to  the  axis  of  x.  During  the  time  dt  this  segment  will  turn  through 
an  infinitely  small  angle,  and  its  length  will  become 


to  the  second  order. 

In  a  latin  paper,  Principia  motus  fluidorum,  Euler  elucidates  the 
problem — then  entirely  novel — of  the  kinematics  of  continuous  media. 
He  calculates  the  form  which  the  elementary  parallelipiped,  whose 
origin  is  Z  and  sides  are  dx,  dy  and  dz,  will  assume  at  the  time  t  +  dt 
because  of  the  motion  of  the  fluid.  He  finds  that  the  volume  becomes 

7   ,    ,   /,    ,     ,  du   ,    T  dv    ,    ,  dw\ 
dxayaz  1 1  +  at  - — \-  dt  - — h  dt  -r-  • 

\  dx  dy  dz/ 

Similarly  the  density,  g,  of  the  fluid  at  Z  becomes,  at  Zr, 


. 

ot  dx  dy  dz 

At  this  point  Euler  expresse  the  conservation  of  the  mass  in  the  course 
of  the  motion.  "  As  the  density  is  reciprocally  proportional  to  the 
volume,  the  quantity  q'  will  be  related  to  q  as  dxdydz  is  related  to 

j  j  j    /i    i     7  du    .     -  dv    ,     .  dw\ 
dxdydz    1  +  dt  —  +  dt  —  +  dt  —    ; 

\  ox  dy  dzj 

whence,  by  carrying   out  the  division,  the  very  remarkable  condition 
which  results  from  the  continuity  of  the  fluid, 

dq   ,       dq          dq  dq         du   ,      dv          dw 

s-t  +  ud-x  +  v^  +  wr2+^x  +  ^  +  ^  =  °' 

This  may  be  written  more  simply  as 


HYDRODYNAMICS 


303 


and,  for  an  incompressible  fluid,  it  reduces  to 

aufoSw;  „ 

dx~^  dy~^  dz 

Euler  then  calculates  the  acceleration  of  the  element  of  fluid  which 
is  at  Z  at  the  instant  t.  He  first  writes  the  components  of  the  velocity 
at  the  point  Z',  to  which  the  point  Z  is  carried  at  the  end  of  the  time 
dz,  in  the  following  form 


.     7    du    ,       7  du    , 
u  +  at  -  —  \-  udt  -  —  k 
dt  dx 

v  +    ... 


du    ,       7  du 

-  —  \-  wdt  — 

dy  dz 


whence  the   acceleration  or  the  increment  of  the  velocity 
du   ,       Su   , 


, 

^  —  r  w  -^~ 
dy  ^      dz 


3v 
St 
dw 


The  pressure  exerts  the  "  accelerative  force,  "  whose  components 


are 


1  dp 

q  dx 


1  dp 

q  dy 


I  dp 

q  dz 


on  the  elementary  mass  of  the  parallelipiped.  Thus  the  equations 
of  motion  of  the  fluid,  to  be  joined  to  the  equation  of  continuity, 
are 

du   ,       du 


^       1  dp       du   ,       du 

p  --  ^-  =  -  --  \-  u-  --       - 
q  dx       dt    '       dx    '       dy 


^r- 
dz 


Euler  was  too  aware  to  misunderstand  the  difficulty  of  the  study  of 
these  equations  of  motion.  Thus  he  wrote — 

"  If  it  does  not  allow  us  to  penetrate  to  a  complete  knowledge 
of  the  motion  of  fluids,  the  reason  for  this  must  not  be  attributed 
to  mechanics  and  the  inadequacy  of  the  known  principles,  for  analysis 
itself  deserts  us  here.  ..." 

Lagrange,  in  this  connection,  wrote — 


304  THE    PRINCIPLES  OF  CLASSICAL   MECHANICS 

"  By  the  discovery  of  Euler  the  whole  mechanics  of  fluids  was 
reduced  to  a  matter  of  analysis  alone,  and  if  the  equations  which 
contain  it  were  integrable,  in  all  cases  the  circumstances  of  the  motion 
and  behaviour  of  a  fluid  moved  by  any  forces  could  be  determined. 
Unfortunately,  they  are  so  difficult  that,  up  to  the  present,  it  has  only 
been  possible  to  succeed  in  very  special  cases.  " 

Without  concerning  ourselves  with  the  particular  problems  which 
he  treats,  we  remark  that  Euler  indicated  the  simplicity  that  results  if 

udx  +  vdy  +  wdz 

is  a  complete  differential.  Much  later  this  was  distinguished  as  the 
case  in  which  a  velocity  potential  existed,  or  the  case  of  irrotational 
motion. 

In  a  third  paper  on  the  motion  of  fluids  l  Eider  draws  attention 
to  a  plane  irrotational  motion  of  an  incompressible  fluid,  which  is 
characterised  by  the  two  conditions 

du       dv  __  n  dv  ___  du 

dx       dy  dx       By 

In  this  connection,  Euler  acknowledges  a  debt  to  d'Alembert  for 
having  conceived  the  device  of  considering  u  —  iv  as  a  function  of 
x  +  iy?  and  u  +  iv  as  a  function  of  x  —  iy.2  (This  was  before  Cauchy 
had  systematised  the  notion  of  analytic  function,  and  long  before 
the  modern  school  of  hydrodynamics  existed.) 

Eider  also  writes,  with  some  hint  of  sarcasm,  "  However  sublime 
may  be  the  investigations  on  fluids  for  which  we  are  indebted  to  MM. 
Bernoulli,  Clairaut  and  d'Alembert,  they  stem  so  naturally  from  our 
two  general  formulae  that  one  cannot  but  admire  this  agreement  of 
their  profound  meditations  with  the  simplicity  of  the  principles  from 
which  I  have  deduced  my  two  equations,  and  to  which  I  was  directly 
led  by  the  first  axioms  of  mechanics.  " 

Just  because  of  the  analytical  difficulties  of  the  general  problem, 
Euler  did  not  misunderstand  the  importance  of  the  part-experimental, 
part-theoretical  considerations  that  were  used  in  hydraulics.  On  the 
contrary,  in  that  he  was  personally  concerned  with  the  Segner  water 
wheel,  had  analysed  the  working  of  turbines  and  had  himself  designed 
a  reaction  turbine,  he  was  a  pioneer  of  modern  technics. 

1  Continuation  des  recherches  sur  la  theorie  du  mouvement  des  fluides,  Memoires  de 
V Academic  de  Berlin,  1755,  p.  316. 

2  This  device  is  used  by  d'ALEMBERT  in  Lis  Essai  sur  la  resistance  des  fluides. 


HYDRODYNAMICS  305 

7.   BORDA   AND   THE    LOSSES    OF   KINETIC   ENERGY   IN   FLUIDS. 

In  this  paragraph  we  shall  follow  a  work  of  Chevalier  de  Borda 
(1733-1799)  called  Memoire  sur  Vecoulement  des  fluides  par  les  orifices 
des  vases.1 

Borda's  analysis  is  based  on  hoth  Daniel  Bernoulli's  hydrodynamics 
and  the  mechanics  of  fluids  which  d'Alembert  had  related  to  his  own 
principle.  At  first  Borda  discusses  problems  of  flow,  and  on  each 
occasion  his  analysis  owes  something  to  Bernoulli  and  d'Alenxbert. 
Notable  among  these  problems  is  the  determination  of  the  contracted 
section — in  this  connection  he  considers  a  re-entrant  nozzle,  where  the 
contracted  section  can  be  calculated  and  turns  out  to  be  equal  to  half 
of  that  of  the  orifice. 

But  the  essential  interest  of  Borda's  study  is  that  he  drew  attention  to 
"  hydro  dynamical  questions  in  which  a  loss  of  living  force  must  be  assumed" 
Such  losses  appear  in  a  tube  which  is  abruptly  enlarged  or  contracted. 
With  a  bold  insight,  Borda  compared  the  phenomenon  which  occurs 
in  the  fluid  to  an  impact  in  which  a  loss  of  kinetic  energy  was  involved — 
that  is,  in  the  language  of  the  time,  to  an  impact  of  hard  bodies. 

First  Borda  establishes  the  following  Lemma,  and  thus  anticipates 
Carnot's  theorem  in  a  special  case. 

"  Lemma.  —  Let  there  be  a  hard  body  a,  whose  velocity  is  u,  which 
hits  another  hard  body  A  whose  velocity  is  V.  It  is  required  to  find 
the  loss  of  living  force  which  occurs  in  the  impact.  2  i  A  r^z 

44  Before  the  impact  the  sum  of  the  living  forces  was . 

After  the  impact  this  sum  has  the  value  ^ 

a  +  A  (an  +  A 


2g      \    a  +  A 

•whence,   by  difference, 

aA      (u  —  F)2 

-~- 


Borda  considers  (see  figure)  the  immersion  of  a  cylindrical  vessel 
into  an  indefinite  fluid  OPQR,  and  seeks  the  motion  which  the  fluid 
will  have  on  entering  the  vessel.  He  starts  from  the  following  con 
sideration. 

"  The  motion  of  the  water  in  the  vessel  can  be  regarded  as  that 
of  a  system  of  hard  bodies  that  interact  in  some  way.  Now  we  know 
that  the  principle  of  living  forces  only  applies  to  the  motion  of  such 

1  Memoires  de  VAcademie  des  Sciences,  1766,  p.  579. 

20 


306 


THE   PRINCIPLES  OF  CLASSICAL   MECHANICS 


todies  when  they  act  on  each  other  by  imperceptible  degrees,  and 
that  there  is  necessarily  a  loss  of  living  force  as  soon  as  one  of  the  bodies 
collides  with  another.  " 


0 


Fig.  102 


In  the  example  with  which  we  are  concerned,  "  the  slice  mopn 
which  enters  the  vessel  at  one  instant,  occupies  the  position  rsqy  at 
the  next  instant.  It  is  clear  that  before  it  occupies  this  position 
the  small  slice  will  have  lost  a  part  of  its  motion  against  the  fluid 
above,  as  if  it  had  been  an  isolated  mass  which  had  been  hit  by  another 
isolated  mass.  But  in  the  case  of  these  two  isolated  masses  there 
would  have  been  a  loss  of  living  force.  Therefore  there  will  also  be 
such  a  loss  in  the  case  that  we  are  discussing.  " 

And  here  is  Borda's  solution,  which  follows  Daniel  Bernoulli's 
method.  Suppose  that  the  fluid  has  travelled  to  EF,  and  that  in 
the  next  instant  it  travel  to  CD.  Put  AE  =  x,  Ag  =  a  and  AB  =  b. 
Let  u  be  the  velocity  of  the  fluid  at  E.  Assume  that  the  living  force 
of  the  fluid  in  the  indefinite  vessel  ROPQ  remains  zero.  Under  these 
conditions,  the  living  force  of  the  fluid  in  the  inner  vessel  can  be  written 

u^bx 


HYDRODYNAMICS  307 

if  the  living  force  of  the   slice  which  enters  the  vessel  is  neglected. 
"  Thus  the  difference  of  the  living  force  of  all  the  fluid  contained 
in  the  vessel  will  be 

u2bdx  -j-  Zbxudu 


Now  while  the  fluid  acquires  this  increment  of  living  force  the  slice 
DCFE,  or  bdx,  is  supposed  to  descend  from  the  height  GjB,  or  a  —  x. 
Therefore,  if  the  principle  of  living  forces  applied  without  restriction, 
it  would  he  true  that 

,  .  ,  7         u2bdx  -4-  2bxudu 

(a  —  X)  bdx  =  --- 

"  But  there  is  a  loss  of  living  force  in  the  whole  of  the  fluid,  which 
arises  from  the  action  of  the  small  slice  rsqy  on  the  fluid  rCDs  which 
is  above  it.  It  is  easy  to  see,  by  the  lemma,  that  if  the  velocity  of 
the  slice  opmn  is  denoted  by  F,  then  this  loss  of  living  force  is 

badx    (V  -  u)2  _          (F~  u)2 
_____  octx  • 

a  +  dx        2g  2g 

"  Therefore,  adding  this  quantity  to  the  second  term  of  the  equation 
above,  the  correct  solution  of  the  problem  is  obtained  — 

u*bdx  +  2bxudu  +  bdx  (V  —  u)2  =  2g  (a  —  x)  bdx. 

u  It  only  remains  to  determine  V.  For  this  purpose  it  is  sufficient 
to  observe  that  the  stream  of  fluid  which  enters  the  vessel  contracts 
in  the  same  way  as  if  it  left  the  vessel  by  the  same  orifice  and  entered 
free  space.  This  must  be  since,  in  both  cases,  the  fluid  which  arrives 
at  the  orifice  is  travelling  in  the  same  directions.  Now  the  loss  of 
living  force  must  be  distributed  from  the  slice  that  has  the  greatest 
velocity  —  that  is,  from  that  which  is  at  the  point  of  greatest  contraction. 

"  Therefore  suppose  that  this  point  is  at  o  and  that  m  is  the  ratio 
of  EF  to  op.  Then  V  =  mu,  whence 

u?dx  -j-  2xudu  +  u2dx  (m  —  I)2  =  2g  (a  —  x)  dx. 

This  equation  is  integrated  by  supposing  that  x  —  e  and  u  =  o  at 
the  beginning  of  the  motion.  " 

Borda  then  repeats  his  argument  and,  this  time,  follows  d'Alembert's 
method. 

"  What  we  have  just  said  of  the  principle  of  conservation  of  living 
forces  is  also  applicable  to  M.  d'Alembert's  principle.  Not  that  the 


308  THE    PRINCIPLES  OF  CLASSICAL   MECHANICS 

latter  principle  is  always  true,  for  there  are  some  instances  in  which 
the  way  it  is  applied  to  the  motion  of  fluids  must  be  somewhat  modified. 
Indeed,  we  have  seen  that  the  slice  rsxy  only  acts  on  the  fluid  above 
in  the  way  that  an  isolated  mass  would  lose  a  part  of  its  motion  to 
another  mass  with  which  it  collided.  Whence  it  follows  that  in  the 
equation  of  equilibrium,  the  accelerating  force  must  not  be  multiplied 

by  the  volume  -  mt  which  the   slice   occupies   at   the   middle   to   the 

Zi 

time  interval,  but  by  the  volume  ot  which  it  occupies  at  the  end  of  this 
interval.  For  the  volume  ot  represents  the  mass  of  the  small  slice 
and  rC  represents  that  of  the  fluid  rCDs.  "  No  purpose  would  be  served 
by  reproducing  the  calculation  which  follows,  which  leads  to  the  same 
result  as  the  analysis  reproduced  above. 

However  bold  it  may  have  been,  Borda's  hypothesis  is  discovered 
to  be  in  satisfactory  agreement  with  experiment. 

"  A  tube  18  lines  in  diameter  and  one  foot  long  was  made  of  very 
uniform  tinplate  whose  edges  were  tapered.  Then,  closing  the  upper 
orifice  with  the  hand,  the  tube  was  plunged  into  a  vessel  filled  with 
water.  It  was  assured  that  the  air  contained  in  the  tube  did  not 
allow  the  water  to  enter  to  the  same  extent  as  if  both  openings  had 
been  free.  Then  the  upper  orifice  of  the  tube  was  opened  and  the 
water  mounted  inside  the  tube  to  a  height  greater  than  its  level  outside. 
The  experiment  was  repeated  several  times  and  the  water  rose  to 
its  peak  which  was  4  pouces  above  the  outside  level.  According  to 
the  calculation  of  M.  Bernoulli,  it  should  have  risen  to  8  pouces.  "  x 

The  ascent  calculated  by  Borda  was  49  %  lines.  He  observed 
an  ascent  of  47%  lines  and  attributed  the  difference  to  the  friction 
of  the  fluid  on  the  walls. 

1  M£moire$  de  I* Academic  des  Sciences,  1766,  p.  147. 


CHAPTER  NINE 


EXPERIMENTS  ON  THE  RESISTANCE  OF  FLUIDS 

(BORDA,  BOSSUT,  DU  BUAT) 
COULOMB  AND  THE  LAWS  OF  FRICTION 


1.  BORDA'S  EXPERIMENTS  AND  NEWTONIAN  THEORIES. 

During  the  same  time  that  the  principles  of  dynamics  were  being 
organised  and  the  foundations  of  hydrodynamics  were  being  developed, 
there  grew  up  a  complete  experimental  approach  that  was  determined 
by  requirements  of  an  essentially  practical  kind.  To  pause  on  this 
remarkable  movement  is  not  to  move  away  from  the  principles  of 
mechanics,  for  here  it  can  be  seen  how  experiment  is  dominant  in 
fields  where  the  theory  is  impotent  before  the  very  complexity  of 
even  the  most  tangible  phenomena. 

We  shall  only  deal  with  some  examples  of  this  experimental  work 
in  mechanics  during  the  XVIIIth  Century.  Besides  being  charac 
teristic,  these  examples  are  ones  in  which  the  origins  of  modern  research 
should  be  sought,  and  in  which  the  modest  methods  deployed  (for 
example,  the  motive  agencies  were  invariably  provided  by  falling 
weights)  were  no  obstacle  to  the  application  of  a  rigorous  experimental 
method. 

But  before  coming  to  these  examples,  it  is  necessary  that  we  should 
describe  some  essays  of  the  theoreticians,  who  had,  indeed,  preceded 
the  experimentalists  by  several  years. 

Newton  had  developed  a  schematic  theory  of  fluids,  which  he 
considered  to  be  formed  of  an  aggregate  of  elastic  particles  which 
repelled  each  other,  were  arranged  at  equal  distances  from  each  other, 
and  were  free.  If  the  density  of  this  aggregate  was  very  small,  Newton 
assumed  that  if  a  solid  moved  in  the  fluid  then  the  parts  of  the  fluid 
which  were  driven  along  by  the  solid  were  displaced  freely,  and  did  not 
communicate  the  motion  which  they  received  to  neighbouring  parts. 

In  this  framework,  Newton  calculated  the  resistance  of  a  fluid 
to  the  translation  of  a  cylinder.  He  found  that  this  resistance  was 


310  THE    PRINCIPLES  OF  CLASSICAL   MECHANICS 

equal  to  the  weight  of  a  cylinder  of  fluid  of  the  same  base  as  the  solid, 
and  whose  height  was  twice  that  from  which  a  heavy  body  would 
have  to  fall  in  order  to  acquire  the  velocity  with  which  the  solid  moved. 

The  resistance  offered  to  the  translation  of  a  sphere,  according 
to  the  same  newtonian  theory,  is  half  the  resistance  which  the  cylinder 
encounters  under  the  same  conditions. 

Jean  Bernoulli  adopted  these  laws  in  the  discussion  of  the  commun 
ication  of  motion  which  he  gave  in  connection  with  the  controversy 
on  living  forces. 

Newton  also  formulated  a  second  theory  on  the  resistance  of  fluids, 
and  applied  this  to  water,  oil  and  mercury.  His  first  theory  was 
only  applied  to  the  resistance  of  air.1 

In  this  second  theory,  particles  of  the  fluid  are  contiguous.  Newton 
compares  the  resistance  to  the  effect  of  the  impact  of  a  stream  of 
fluid  on  a  circular  surface,  the  stream  being  imagined  to  leave  a  cylin 
drical  vessel  through  a  horizontal  orifice.  He  passes  to  the  limit 
by  infinitely  increasing  the  capacity  of  the  vessel,  and  also  the  dimensions 
of  the  orifice,  in  order  to  simulate  the  conditions  of  an  indefinite  fluid. 
He  then  substitutes  the  motion  of  the  circular  surface  for  that  of  the 
fluid  in  the  first  model  of  impact. 

Given  this,  Newton  calculates  the  resistance  offered  to  the  trans 
lation  of  a  cylinder  and  finds  that  the  resistance  is  equal  to  the  weight 
of  a  cylinder  of  fluid  whose  base  is  the  same  as  that  of  the  solid  and 
whose  height  is  half  that  from  which  a  heavy  body  would  have  to  fall 
in  order  to  acquire  the  velocity  with  which  the  solid  moves  in  the 
fluid.  This  resistance  is  four  times  smaller  than  that  provided  by 
the  first  theory. 

Further,  in  the  second  theory  the  length  of  the  moving  cylinder 
does  not  affect  the  result,  for  only  its  base  is  exposed  to  the  impact 
of  the  fluid.  Under  these  circumstances  the  resistance  offered  to  the 
translation  of  a  sphere  is  equal  to  that  which  would  be  offered  to  the 
translation  of  a  cylinder  circumscribed  about  the  sphere.  This  result 
is  half  that  provided  by  the  first  theory. 

The  second  newtonian  theory  is  applicable  to  the  oblique  impact 
of  a  stream  of  fluid  on  a  plane  wall.  Under  these  conditions,  it  leads 
to  a  resistance  which  is  proportional  to  both  the  square  of  the  velocity 
and  the  square  of  the  sine  of  the  angle  of  incidence.  These  were  the 
proportions  which  the  experimenters  tried  to  verify. 

We  also  add  that  Daniel  Bernoulli,  although  he  did  not  offer  an 
alternative  theory,  had  already  remarked  on  considerable  differences 

1  In  fact  this  theory  goes  back  to  HUYGHENS  (1669),  MARIOTTE  (1684)  and  PARTIES 
(1671). 


THE    RESISTANCE    OF    FLUIDS 


311 


between  the  newtonian  laws  and  experiment.1  Moreover,  he  legitim 
ately  emphasised  that  it  was  necessary  to  distinguish  between  the 
impact  of  a  fluid  on  a  wall  and  the  impact  of  a  fluid  on  a  completely 
immersed  plane.2  Finally,  we  recall  that  d'Alembert,  in  his  early 
work  on  the  resistance  of  fluids,  also  calculated  the  impact  of  a  moveable 
surface  on  an  infinity  of  small  elastic  balls  which  represented  a  fluid.3 

With  this  in  mind,  we  come  to  the  experiments  of  Chevalier  de  Borda. 

In  the  first  place,  Borda  studied  the  resistance  of  air.4  By  means 
of  a  driving  weight  he  made  a  flywheel  rotate  and  attached  plane 
surfaces  of  different  shapes  to  the  circumference.  He  took  care  to 
correct  the  results  for  the  friction  of  the  flywheel  and  to  confine  the 
observations  to  a  period  of  uniform  motion,  when  a  steady  state  had 
been  established. 

These  are  Borda's  conclusions. 

1)  The  total  resistance  of  the  air  cannot  be  calculated  as  the  sum 
of  the  partial  resistances  of  each  of  its  elements.     For  example,  the 
resistance  of  a  circle  is  not  the  sum  of  the  resistances  of  two  semicircles. 
This  conclusion  is  very  important — it  shoes  that  the  resistance  is  a 
phenomenon  which  behaves  integrally,  and  also  makes  it  clear  that 
the  resistance  cannot  be  obtained  by  an  integration  which  depends 
on  a  simple  elementary  law. 

2)  The  aggregate  resistance  is  proportional  to  the  square  of  the 
velocity  and  the  sine  of  the  angle  of  incidence  (not  to  the  square  of 
this   sine). 


Fig.  103 

As  far  as  the  resistance  of  water  is  concerned,  Borda  confines  him 
self,  in  this  first  paper,  to  the  verification  of  the  proportionality  to 
the  square  of  the  velocity. 

1  Mtmoires  de  Petersburg,  Vol.  II,  1727. 

2  Ibid.,  Vol.  VIII,  1741. 

3  See  above,  p.  295. 

4  Memoires  de  V Academic  des  Sciences,  1763,  p.  358. 


312  THE   PRINCIPLES  OF  CLASSICAL   MECHANICS 

Borda  returned  to  tlxe  resistance  of  fluids  in  a  paper  dated  1767.1 

He  worked  with  a  circular  vessel  12  feet  in  diameter.  By  means 
of  driving  weights  varying  from  4  ounces  to  8  pounds,  he  made  a 
sphere  of  59  lines  diameter  move  through  the  water.  The  sphere 
was  made  of  two  equal  parts,  which  could  be  joined  together  or  separ 
ated  as  desired. 

When  working  with  one  hemisphere,  Borda  allowed  it  to  present 
either  the  section  of  a  great  circle  or  the  convex  part,  to  the  fluid. 

Borda  took  care  to  allow  for  "  the  friction  and  the  impact  of  the 
air  on  the  flywheel  "  by  making  the  apparatus  rotate  freely  without 
the  sphere. 

He  verified  that  the  resistance  was  very  accurately  proportional 
to  the  square  of  the  velocity.  In  addition,  he  established  that  the 
resistance  of  the  hemisphere  was  nearly  independent  of  the  surface 
that  was  presented  to  the  fluid.  From  this  he  concluded  that  "  at 
these  small  velocities.,  the  forward  part  of  the  body  is  the  only  one  ichich 
has  resistance.  " 

Borda  next  turned  his  attention  to  the  absolute  magnitude  of  the 
resistance,  and  compared  the  values  observed  with  those  calculated 
from  what  we  have  called  the  second  newtonian  theory  (Principia, 
Book  II,  Proposition  XXXVIII).  He  found  that  the  resistance  of  the 
hemisphere  when  it  offered  a  section  of  a  great  circle  to  the  fluid  was 
2%  times  as  great  as  the  resistance  of  the  whole  sphere,  itself  accur 
ately  equal  to  the  resistance  of  the  hemisphere  when  this  offered  its 
convex  side  to  the  fluid.  Now,  according  to  the  newtonian  theory,  the 
first  resistance  is  twice  the  second.  The  disagreement  is  evident. 

Similarly,  Borda  determined  the  oblique  resistance.  He  established, 
exactly  as  in  his  experiments  in  air,  that  the  law  of  the  square  of  the 
sine  was  not  true,  and  even  declared  "that  when  the  angles  of  incidence 
are  small  the  resistance  does  not  decrease  as  much  as  the  simple  sine.  " 

Borda  also  studied  the  influence  of  the  depth  on  the  resistance 
in  water.  He  established  that  the  resistance  decreased  with  the 
depth,  and  that,  at  the  surface,  it  increased  more  rapidly  than  the 
square  of  the  velocity.  In  this  connection,  he  attempted  an  explan 
ation  which  was  only  half  convincing,  by  falling  back  on  his  own 
theory  of  the  losses  of  living  force  in  fluids.2 

"  It  is  clear  that  when  the  sphere  is  only  6  pouces  below  the  surface 
it  does  not  impart  such  great  velocities  to  the  neighbouring  parts 
as  when  it  moves  in  the  surface  of  the  water.  For  in  the  first  case 
the  fluid  is  free  to  run  round  the  whole  circumference  of  the  sphere 

1  Memoires  de  VAcademie  des  Sciences,  1767,  p.  495 

2  See  above,  p.  305. 


THE    RESISTANCE    OF   FLUIDS  313 

while  in  the  second,  it  cannot  escape  along  the  upper  part  of  the  sphere. 
From  which  it  follows  that  in  the  first  instance  the  fluid  neither  gains 
nor  loses  as  great  a  quantity  of  living  forces  as  in  the  second.  " 

Borda  then  worked  with  a  model  ABCD 
in  which  AH  =  HD  =  6  pouces  and  J3C  = 
4  pouces. 

The  difference  between  the  resistance 
when  the  side  A  (angle  BAC),  and  then 
the  side  D  (two  arcs  of  circles,  BD  and 
DC,  with  centres  on  J3C),  were  offered  to  Pig.  104 

the   fluid    was   negligible — the   newtonian 
theory    predicted     a    ratio     of    28     to     15     for     these     resistances. 

Borda's  general  conclusion  was  that  the  newtonian  theory  could 
not  account  for  the  resistances  of  fluids.  "  The  ordinary  theory  of 
the  impact  of  fluids  only  gives  relationships  which  are  absolutely 
false  and,  consequently,  it  would  be  useless  and  even  dangerous  to 
wish  to  apply  this  theory  to  the  craft  of  the  construction  of  ships.  " 

2.  THE  ABBE  BOSSUT'S  EXPERIMENTS. 

In  1775  Turgot  asked  the  Academie  des  Sciences  "  to  examine 
means  of  improving  navigation  in  the  Realm.  "  A  committee  consisting 
of  d'Alembert,  Condorcet  and  the  Abbe  Bossut  (as  secretary)  immediat 
ely  took  up  the  investigation  and,  between  July  and  September,  1775, 
conducted  numerous  experiments  "  on  a  large  stretch  of  water  in  the 
grounds  of  the  Military  College.  "  They  secured  the  cooperation  of 
the  mathematicians  attached  to  that  College,  including  Legendre  and 
Monge. 

The  committee  reported  to  the  Academie  des  Sciences  on  April 
17th,  1776,  and  this  report,  Nouvelles  experiences  sur  la  resistance  des 
fluides  was  published  at  Paris  (Jombert)  in  1777,  under  the  names 
of  the  three  members  of  the  committee. 

The  experimental  method  is  referred  to  in  the  following  terms. 
"  To  ask  questions  of  nature  by  doing  experiments  is  a  very  delicate 
matter.  In  vain  do  you  assemble  the  facts  if  these  have  no  relation 
to  each  other  ;  if  they  appear  in  an  equivocal  form ;  if,  when  they 
are  produced  by  different  causes,  you  are  unable  to  assign  and  distin 
guish  the  particular  effects  of  these  causes  with  a  certain  precision.  .  .  . 
Do  not  heed  the  limited  experimenter,  the  one  who  lacks  principles  ; 
guided  by  an  unreasoning  method,  he  often  shows  us  the  same  fact 
in  different  guises — of  necessity,  and  perhaps  without  recognising 
this  himself;  or  he  gathers  at  random  several  facts  whose  differences 


314  THE   PRINCIPLES  OF  CLASSICAL   MECHANICS 

he  is  unable  to  explain.  A  science  without  reasoning  does  not  exist  or, 
what  comes  to  the  same  thing,  a  science  without  theory  does  not  exist.  "  * 

Bossut  explicitly  distinguished  between  the  resistance  of  fluids 
that  were  indefinitely  extended  (a  ship  on  the  sea  or  on  wide  and 
deep  rivers)  and  the  resistance  in  narrow  channels  (shallow  or  narrow 
rivers  and  canals). 

Borda's  experiments  were  conducted  in  fluids  that  were,  for  practical 
purposes,  indefinitely  extended.  On  the  other  hand,  in  order  to  study 
the  effect  of  the  depth  of  immersion,  Franklin  had  worked  on  a  small 
scale  with  a  canal  and  a  model  of  a  ship  which  was  6  pouces  long,  and 
2%  pouces  wide.2 

The  basin  at  the  Military  College  was  100  feet  long  and  53  feet 
wide  at  the  centre,  its  maximum  depth  being  6%  feet.  A  weight  hung 
over  a  pulley  assured  the  traction  of  the  model,  which  was  equiped 
with  a  rudder  in  order  that  its  motion  might  be  determinate. 


Bossut's  Model  No.  1. 
Fig.  105 

Bossut  used  twelve  different  models  of  ships  and  carried  out  a 
total  of  about  300  trials,  of  which  about  200  were  in  an  effectively 
indefinite  fluid  and  the  remainder  in  an  artificially  constructed  channel 
whose  depth  and  width  were  variable  at  will. 

When  he  compared  the  experimental  results  with  the  second  new- 
tonian  theory,  Bossut  came  to  the  following  conclusions. 

1)  On  a  given  surface,  and  at  different  velocities,  the  resistance 
is  "  approximately  in  the  square  ratio,  just  as  much  for  oblique  impacts 
as  for  direct  impacts.     More  accurately,  the  resistance  increases   in 
a  greater  ratio  than  the  square.  "     He  gives  the  following  explanation 
of  this   fact.     "  The   fluid  has   greater   difficulty  in   deflecting  itself 
when  the  velocity  increases — it  piles  up  in  front  of  the  prow  and  is 
lowered  near  the  stern.  "  3 

2)  "  For  surfaces  which  are  equally  immersed  in  the   fluid  and 
only  different  in  respect  of  their  width,  the  resistance  sensibly  follows 

1  Nouvelles  experiences  sur  la  resistance  des  fluides,  p.  5. 

2  CEuvres  completes  de  Franklin,  Vol.  II,  p.  237. 

3  BOSSUT,  op.  cu.,  p.  147. 


THE  RESISTANCE  OF  FLUIDS  315 

the  ratio  of  the  surfaces.  .  .  .  More  precisely,  it  increases  in  a  ratio 
which  is  a  little  greater  than  that  of  the  extent  of  the  surfaces.  "  1 

"  The  resistance  of  bodies  which  are  entirely  submerged  is  a  little 
less  than  that  of  bodies  which  are  only  partly  submerged.  "  2 

3)  "  The  law  of  the  square  of  the  sine  is  less  justified  when  the  angles 
are  very  small.  "  3  In  order  to  express  the  results  of  these  experiments, 
the  Abbe  Bossut  chose  a  provisional  law  of  the  form 

sin"£ 

where  i  is  the  angle  of  incidence.  He  found  that  the  exponent  n  varied 
from  0.66  to  1.79,  according  to  the  model  studied.  This  led  him 
to  conclude — 

"  The  resistances  which  occur  in  oblique  impacts  cannot  be  explained 
by  the  theory  of  resistances  by  introducing,  instead  of  the  square, 
some  other  power  of  the  sine  of  the  angle  of  incidence  in  the  expression 
for  the  resistance.  "  4 

In  order  to  determine  the  magnitude  of  the  resistance  of  water, 
Bossut  made  two  corrections.  The  first  depended  on  the  friction 
of  the  pulley  which  supported  the  cable  and  the  motive  weight — he 
measured  this  friction  by  varying  the  motive  weight.  The  second 
correction  arose  because  of  "  the  impact  of  the  air  "  on  the  model. 
Indeed,  to  the  author,  resistance  was  an  impact  phenomenon.  He  was 
guided  throughout  by  the  second  newtonian  theory.  In  order  to  eli 
minate  the  impact  of  the  air,  Bossut  measured  the  surface  of  the  model 
which  was  offered  to  the  impact  of  air,  and  assumed  that  the  impacts  of 
the  water  and  the  air  on  the  model  were  respectively  "in  compound 
proportion  to  the  impacted  surfaces  and  the  densities  of  the  two  fluids." 

Having  made  these  two  corrections,  Bossut  concluded — 

"  The  resistance  perpendicular  to  a  plane  surface  in  an  indefinite 
fluid  is  equal  to  the  weight  of  a  column  of  fluid  having  the  impacted 
surface  as  its  base  and  whose  height  is  that  which  corresponds  to  the 
velocity  with  which  the  percussion  occurs.  "  5 

Bossut  tried  to  analyse  further  the  phenomenon  of  resistance  ;  he 
sought  to  emphasize  the  part  played  by  the  "  tenacity  "  of  the  fluid  and 
the  "  friction  caused  along  the  length  of  the  boat  by  the  water.  "  From 
this  somewhat  arbitrary  decomposition,  he  felt  justified  in  drawing  the 
following  conclusions  : 

"  We  have  observed  that  as  soon  as  the  friction  is  overcome,  the 

1  BOSSUT,  op.  cit.,  p.  152. 

2  Ibid.,  p.  157. 
8  Ibid.,  p.  163. 

4  Ibid.,  p.  164. 

5  Ibid.,  p.  173. 


316  THE   PRINCIPLES   OF  CLASSICAL   MECHANICS 

slightest  force  sets  the  hoat  in  motion.  From  which  we  have  concluded 
that  the  tenacity  of  the  water  is  extremely  small  and  that  this  resistance 
must  be  considered  absolutely  nil  in  comparison  with  that  caused  by 
inertia.  The  same  applies  to  the  friction  of  the  water  along  the  sides 
and  bottom  of  the  boat.  This  friction  is  very  slight  and  its  effect  cannot 
be  distinguished  from  that  of  the  pulleys  or  of  the  resistance  of  the  air.  "  x 
Again  Bossut  noted  the  resistance  in  a  narrow  canal,  superior  to  the 
resistance  in  an  unlimited  fluid,  and  he  underlined  the  influence  of  the 
transversal  dimensions  and  of  the  form  of  the  vessel  used  for  comparison. 
For  the  construction  itself  of  the  canals,  his  paper  is  limited  to  cautious 
generalities  :  the  canal  should  be  as  large  and  as  deep  as  possible, 
"  without  nevertheless  going  to  superfluous  expense  "  ;  subterranean 
canals  should  be  avoided  unless  local  circumstances  make  their  use 
indispensable.  Indeed,  concludes  this  sagacious  rapporteur,  "  a  canal 
is  an  object  of  utility  and  not  an  instrument  for  ostentation.  " 

3.  Du  BUAT  (1734-1809).  HYDRAULICS  AND  THE  RESISTANCE  OF  FLUIDS. 

Du  Buat  began  by  directing  the  construction  of  fortifications  and  on 
this  occasion  was  the  promotor  of  *  geometric  cotee  '.  He  later  devoted 
himself  to  hydraulics,  as  "  Captain  of  the  infantry,  engineer  to  the  King." 

The  Principes  tfhydraulique,  the  first  edition  of  which  is  dated  1779, 
deals  with  "  the  motion  of  water  in  rivers,  canals  and  conduits  ;  the 
origin  of  rivers  and  the  formation  of  their  beds  ;  the  effect  of  locks, 
bridges  and  reservoirs  2  ;  of  the  impact  of  water  ;  and  of  navigation  on 
rivers  as  well  as  on  narrow  canals.  " 

Du  Buat  wrote,  "  there  is  no  argument  which  can  be  used  to  apply 
the  formulae  for  flow  through  orifices  to  the  uniform  flow  of  a  river, 
which  can  only  owe  the  velocity  with  which  it  moves  to  the  slope 
of  its  bed,  taken  at  the  surface  of  the  current.  " 

Gravity  is,  on  both  cases,  certainly  the  cause  of  the  motion.  "  I 
therefore  set  out  to  consider  whether,  if  water  was  perfectly  fluid  and 
ran  in  the  part  of  a  bed  which  provided  no  resistance,  it  would  accelerate 
its  motion  like  bodies  which  slide  on  an  inclined  plane.  .  .  .  Since 
it  is  not  so,  there  exists  some  obstacle  which  prevents  the  accelerating 
force  from  imparting  fresh  degrees  of  velocity  to  it.  Now,  of  what 
can  this  obstacle  consist,  except  the  friction  of  the  water  against  the 
walls  of  the  bed  and  the  viscosity  of  the  fluid  ?  " 

And  Du  Buat  stated  this  principle —  "  When  water  runs  uniformly 
in  some  bed,  the  force  which  is  necessary  to  make  it  run  is  equal  to 

1  BOSSUT,  op.  cit.,  p.  173. 

2  Read  "  weirs.  " 


THE    RESISTANCE    OF   FLUIDS  317 

the  sum  of  the  resistances  to  which  it  is  subject,  whether  they  are  due 
to  its  own  viscosity  or  to  the  friction  of  the  hed.  " 

In  the  1786  edition  of  his  Traite,  Du  Buat  amends  this  statement 
and  no  longer  speaks  of  the  viscosity,  but  only  of  the  resistance  of 
the  bed  or  the  containing  walls.  The  viscosity  only  enters  indirectly, 
44  in  order  to  communicate  the  retardation  due  to  the  walls,  step  by 
step,  to  those  parts  of  the  fluid  which  are  not  in  contact  with  them.  " 
This  effect  only  influences  "  the  relation  between  the  mean  velocity 
and  that  which  is  possessed  by  the  fluid  against  the  walls.  " 

In  canals  of  circular  or  rectangular  section,  Du  Buat  introduced 
the  notion  of  mean  radius  (the  ratio  of  the  area  of  the  cross-section 
to  the  length  of  the  perimeter  in  contact  with  the  fluid)  and  evaluated 
the  resistance  of  the  walls  to  unit  length  of  the  current  by  the  product 
of  this  radius  and  the  friction  on  unit  surface.  He  assumed  that  the 
resistance  of  the  bottom  was  proportional  to  the  square  of  the  velocity 
of  the  current,  and  likened  it  to  the  impact  of  the  water  on  the  irre 
gularities  on  the  bottom. 

Du  Buat  did  not  confine  himself  to  this  theoretical  outline  but, 
like  the  Abbe  Bossut,  he  sought  experimental  confirmation.  The 
second  edition  of  his  Traite  is  concerned  with  these  experiments. 

Du  Buat  verified  that  the  friction  of  fluids  was  independent  of 
their  pressure.  This  he  did  by  making  water  oscillate  in  two  siphons 
of  very  different  depth.  He  investigated  the  friction  of  fluids  on  different 
materials  (glass,  lead  and  tin)  and,  having  observed  that  this  friction 
was  always  the  same,  he  assumed  that  the  water  "  itself  prepares  the 
surface  on  wliich  it  runs  "  by  wetting  the  pores  and  cavities  as  a  varnish 
does.  He  went  further,  and  even  held  that  the  resistance  of  the  walls 
did  not  depend  on  their  roughness — a  conclusion  that  was  very  far 
from  being  correct. 

In  order  to  obtain  the  resistance  of  the  walls,  Du  Buat  worked 
with  an  artificial  canal  of  oak  planks,  whose  section  could  be  varied 
in  shape  and  size.  He  also  worked  with  pipes  of  tinplate  or  glass 
of  very  different  diameters.  He  discovered  that  the  resistance  of 
the  walls  was  in  a  smaller  ratio  than  the  square  of  the  velocity  x  and 

1  DE  PHONY  advocated  a  formula  for  the  resistance  which  had  the  form  av  -f  bv2. 
Much  later,  after  having  observed  the  oscillations  of  a  circular  plate  in  a  fluid  medium, 
COULOMB  was  to  say,  "  There  must  be  two  kinds  of  resistance.  One,  due  to  the  coher 
ence  of  the  molecules  which  are  separated  from  each  other  in  a  given  time,  is  propor 
tional  to  the  number  of  these  molecules  and,  consequently,  to  the  velocity.  The  other, 
due  to  the  inertia  of  the  molecules  which  are  stopped  by  the  roughnesses  with  which 
they  collide,  is  proportional  to  both  their  number  and  their  velocity  and,  consequently, 
to  the  square  of  their  velocity.  "  COULOMB  was,  before  STOKES,  the  first  to  hold  that 
the  velocity  of  a  viscous  fluid  relative  to  a  solid  was  nothing  at  the  surface  of  contact, 
and  that  it  then  varied  continuously  in  the  fluid. 


318  THE   PRINCIPLES   OF  CLASSICAL   MECHANICS 

gave  an  empirical  formula  for  this  resistance  which  was  only  surpassed 
in  accuracy  by  those  of  Darcy  (1857)  and  Bazin  (1869). 

Du  Buat  then  turned  to  the  empirical  relationships  between  the 
mean  velocity,  the  velocity  at  the  centre  of  the  surface  and  the  velocity 
at  the  centre  of  the  bed.  To  account  for  the  resistance  due  to  bends, 
Du  Buat  assumed  a  series  of  impacts  on  the  banks,  and  expressed 
the  resistance  as  a  number  proportional  to  the  square  of  the  mean 
velocity,  the  square  of  the  sine  of  the  angle  of  incidence  and  the  number 
of  "  ricochets.  "  He  applied  his  empirical  formula  to  the  eddies  and 
local  variations  of  level  which  are  found  upstream  from  barrages  and 
narrows  by  considering  small  consecutive  lengths  of  the  current. 

Du  Buat  also  treated  the  decrease  of  the  slope,  and  the  increase 
of  the  depth,  from  the  source  of  a  river  to  its  mouth.  He  took  account 
of  tributaries,  temporary  and  periodic  floods,  changes  of  course,  the 
retarding  effect  of  the  wind  and  even  the  influence  of  the  weeds  which 
grew  in  the  bottom. 

In  order  to  ascertain  the  resistance  of  fluids  to  the  translation  of  a 
solid,,  Du  Buat  exposed  a  tinplate  box  to  the  current.  The  box  was  either 
cylindrical  or  in  the  form  of  a  parallelipiped  whose  edges  were  parallel 
to  the  flow  lines.  The  boxes  were  provided  with  holes  which  could 
be  opened  or  closed  at  will.  A  float  allowed  the  difference  of  the 
levels,  outside  and  inside  the  box,  to  be  measured,  and  thus  the  pressures 
at  different  points  of  the  surface  of  the  box  to  be  estimated.  In  this 
way  Du  Buat  showed  the  existence  of  an  over-pressure  at  the  front 
(with  respect  to  the  previously  existing  state,  in  which  the  level  was 
uniform)  and  a  "  non-pressure,  "  or  suction  "  at  the  back  acting  in 
the  same  direction  as  the  over-pressure.  " 

The  total  observable  resistance  corresponds  to  the  sum  of  these 
two  effects.  Du  Buat  measured  them  separately,  and  showed  that  the 
over-pressure  was  approximately  the  same  for  a  thin  plate,  for  a  cube 
and  for  a  parallelipiped.  On  the  other  hand,  the  "  non-pressure  " 
decreased  rapidly  when  the  solid  became  relatively  longer. 

Du  Buat  found  that  the  resisting  force  of  a  fluid  mass  to  a  solid  in 
translation  was  less  than  the  resistance  of  the  solid  at  rest  to  the  moving 
fluid,  if  the  relative  velocity  was  the  same  in  both  cases.  This  is  ex 
plained  by  the  fact  that  he  worked  on  a  limited  fluid  mass.  Du  Buat 
then  set  out  to  measure  the  amount  of  the  fluid  which  accompanied 
a  solid  in  its  motion  through  a  practically  indefinite  fluid.  He  made 
a  solid  oscillate,  like  a  pendulum  in  the  fluid,  and  studied  the  variation 
of  the  amplitude  of  small  oscillations — a  consequence  of  the  decrease 
of  the  weight  of  the  solid  body  due  to  the  upthrust  of  the  fluid,  and 
the  increase  of  its  mass  due  to  the  mass  a  fluid  carried  along.  If  p 


THE    RESISTANCE    OF   FLUIDS  319 

is  the  weight  of  the  oscillating  body  (weighed  in  the  fluid),  P  the  weight 
of  fluid  displaced,  nP  the  sum  of  the  weights  of  the  fluid  displaced 
and  the  fluid  carried  along,  /  the  length  of  the  pendulum  and  a  the  length 
of  an  isochronous  pendulum  in  the  vacuum,  then 


I 


T_  P    I  a  1\ 

whence     n  =  4r  (  — II- 


a        p+nP      "        P  U 

Indeed,  in  the  fluid 

P  +  nP  pg 

*L J  -,   —_    n        ^T        «,   =    £-2_ 


—  y  =        ,-~n 

g  r        p+nP 


and  also 


Du  Buat  estabKshed,  by  working  in  water  with  metallic  bodies 
and  in  air  with  distended  balloons,  that  the  amount  of  fluid  carried 
along  by  solids  was  approximately  proportional  to  the  resistance 
obtained  by  other  methods.  Further,  he  suggested  extending  the 
measurement  of  oscillations  in  order  to  determine  the  resistance  of  fluids, 
by  working  with  pendulums  consisting  of  long  columns,  so  that  the 
curvature  of  the  trajectory  might  be  a  minimum. 

From  all  these  investigations,  which  place  Du  Buat  among  the 
greatest  experimenters  of  his  time,  the  author  concludes  that  he  has 
"  not  done  much  more  than  destroy  the  old  theoretical  structure, " 
and  he  appealed  for  more  experiments,  in  the  hope  that  a  more  correct 
theory  might  emerge  from  them. 

4.  COULOMB'S  WORK  ON  FRICTION. 

Coulomb  was  not  the  first  to  make  experiments  on  the  friction 
of  sliding  and  the  stiffness  of  ropes. 

Amontons,  in  1699 1,  had  stated  that  the  friction  was  proportional 
to  the  mutual  pressure  of  the  parts  in  contact.  Muschenbroek  intro 
duced  the  amount  of  the  area  of  contact.  De  Camus,  in  a  Traite  des 
forces  mouvantes,  and  D^saguillers  in  a  Cours  de  physique,  remarked  that 
the  friction  at  rest  was  much  greater  than  the  friction  in  motion. 

In  connection  with  the  stiffness  of  ropes,  Amontons  showed  that 
the  force  necessary  to  bend  a  rope  round  a  cylinder  was  inversely 
proportional  to  the  radius  of  the  cylinder  and  directly  proportional 
to  the  tension  and  the  diameter  of  the  rope. 

1  Memoires  de  PAcademie  des  Sciences,  1699. 


320  THE    PRINCIPLES  OF   CLASSICAL   MECHANICS 

In  1781  the  Academie  des  Sciences  chose  the  subject  of  the  laws 
of  friction  and  the  stiffness  of  ropes  for  a  competition,  asking  for  a 
return  "  to  new  experiments,  made  on  a  large  scale  and  applicable 
to  machines  valuable  to  the  Navy,  such  as  the  pulley,  the  capstan 
and  the  inclined  plane.  " 

Coulomb,  who  was  then  senior  captain  of  the  Royal  Corps  of  En 
gineers,  won  the  prize  with  his  Theorie  des  machines  simples  en  ayant 
egard  au  frottement  et  a  la  roideur  des  cordages.1 

In  the  frontispiece  of  his  paper,  Coulomb  quotes  this  saying  of 
Montaigne —  "  Reason  has  so  many  forms  that  we  do  not  know  which 
to  choose — Experiment  has  no  fewer  "  (Essais,  Book  III,  Chapter 
XIII).  In  fact  Coulomb's  work  is  a  model  of  experimental  analysis, 
carried  out  with  precision  and  exemplary  detail,  and  from  which  he 
obtained  a  theory  applicable  to  machines. 

The  parameters  which  Coulomb  used  in  his  study  of  friction  were 
the  following —  the  nature  of  the  surfaces  in  contact  and  of  their 
coatings ;  the  pressure  to  which  the  surfaces  are  subject ;  the  extent 
of  the  surfaces  ;  the  time  that  has  passed  since  the  surfaces  were  placed 
in  contact ;  the  greater  or  lesser  velocity  of  the  planes  in  contact  ; 
and,  incidentally,  the  humid  or  dry  condition  of  the  atmosphere. 

He  described  his  apparatus  in  great  detail  and,  for  example,  mention 
ed  "  a  plank  of  oak,  finished  with  a  trying-plane  and  polished  with 
seal-skin.  "  He  studied  the  friction  of  oak  on  oak,  "  seasoned,  along 
the  grain  of  the  wood,  with  as  high  a  degree  of  polish  as  skill  could 
achieve. "  All  the  result  obtained  were  recorded,  experiment  by 
experiment,  with  the  rigor  of  an  official  report. 

He  first  studied  the  friction  of  sliding  between  two  pieces  of  seasoned 
wood  (oak  on  oak,  oak  on  fir,  fir  on  fir,  elm  on  elm).  He  then  studied 
the  friction  between  wood  and  metals,  between  metals  with  or  without 
coatings,  etc.  .  . . 

By  way  of  an  example,  here  is  a  summary  of  some  of  his  conclusions. 

"  1.  The  friction  of  wood  sliding,  in  the  dry  state,  on  wood  opposes 
a  resistance  proportional  to  the  pressures  after  a  sufficient  period 
of  rest;  in  the  first  moments  of  rest  this  resistance  increases  appreciably, 
but  after  some  minutes  it  usually  reaches  its  maximum  and  its  limit. 

"  2.  When  wood  slides,  in  the  dry  state,  on  wood,  with  any  velocity, 
the  friction  is  once  more  proportional  to  the  pressures  but  its  intensity 
is  much  less  that  which  is  discovered  on  detaching  the  surfaces  after 
some  moments  of  rest. 

"  3.  The  friction  of  metals  sliding  on  metals,  without   coatings, 

1  Memoires  des  Savants  etrangers,  Vol.  X. 


THE    RESISTANCE    OF   FLUIDS  321 

is  similarly  proportional  to  the  pressures  but  its  intensity  is  the  same 
whether  the  surfaces  are  detached  after  some  moments  of  rest,  or 
whether  they  are  forced  into  some  uniform  velocity. 

"  4.  Heterogeneous  surfaces,  such  as  wood  or  metals,  sliding  upon 
each  other  without  coatings,  provide,  in  their  friction,  very  different 
results  from  the  preceding  ones.  For  the  intensity  of  their  friction, 
relatively  to  a  time  of  rest,  increases  slowly  and  only  reaches  its  limit 
after  four  or  five  days,  or  even  more.  ,  .  .  Here  the  friction  increases 
very  appreciably  as  the  velocities  are  increased,  so  that  the  friction 
increases  approximately  in  an  arithmetic  progression  when  the  velo 
cities  increase  according  to  a  geometric  progression.  " 

The  most  debatable  part  of  Coulomb's  paper  is  that  in  which  he 
attemps  to  construct  a  model  of  the  production  of  friction. 

46  The  friction  can  only  arise  from  the  engaging  of  the  projections 
from  the  two  surfaces,  and  coherence  can  only  affect  it  a  little.  .  .  . 
The  fibres  of  wood  engage  in  each  other  as  the  hairs  of  two  brushes 
do  ;  they  bend  until  they  are  touching  without,  however,  disengaging  ; 
in  this  position  the  fibres  which  are  touching  each  other  cannot  bed 
themselves  down  further,  and  the  angle  of  their  inclination,  depending 
on  the  thickness  of  the  fibres,  will  be  the  same  under  all  degrees  of 
pressure.  Therefore  a  force  proportional  to  the  pressure  will  be  necess 
ary  for  the  fibres  to  be  able  to  disengage.  " 

At  first  Coulomb  used  the  same  arrangement  as  Amontons  for  the 
investigation  of  the  stiffness  of  ropes.  Later  he  developed  a  new 
one  which  allowed  him  to  work  with  more  industrial  cables,  namely, 
"  ropes  of  three  untarred  strands.  "  He  summarised  the  effect  of 
the  stiffness  of  ropes  by  means  of  the  formula 

A  +  BT 
R 

where  A  =  hrq,  B  =  h'r**  where  R  is  the  radius  of  the  pulley,  r  the 
radius  and  T  the  tension  of  the  rope.  The  exponents  q  and  ^  are 
approximately  equal. 

The  mechanics  of  friction  was  still  a  very  skeletal  one  in  Coulomb's 
paper.  Coulomb  assumes  that,  in  order  to  draw  a  weight  P  along 
a  horizontal  plane,  it  is  necessary  to  deploy  a  force 

T=  A  +  — 
P 

In  this  formula,  A  is  a  small  constant  depending  on  the  "  coherence  " 
of  the  surfaces  and  JJL  is  a  coefficient  (the  reciprocal  of  the  coefficient 


322 


THE   PRINCIPLES   OF  CLASSICAL   MECHANICS 


of  friction  which  is  now  commonly  used)  depending  on  the  nature  of 
the  surfaces. 

Turning  his  attention  to  the  observations  made  of  the  launching 
of  ships  at  the  port  of  Rochefort  in  1779,  Coulomb  calculated  the 


Fig.  106 


force  necessary  to  hold  a  body  on  an  inclined  plane.     He  obtained 
the  result  that 

__  AJLL  +  P  (cos  7i  +  ju,  sin  n) 
H  cos  m  +  sin  m 

where  n  is  the  inclination  of  the  plane  and  m  the  angle  between  the  force 
T  and  the  plane  BC.     From  this  he  easily  deduced  that  T  is  a  minimum 

«  cos  m 

lor  u  =  — . 

sin  m 

The  mechanics  of  friction  was  born  of  some  experiments  in  physics 
in  the  XVIIth  Century  and  then,  for  an  essentially  practical  purpose, 
was  systematised  by  Coulomb.  But,  at  the  time,  it  remained  linked 
to  the  common  practice  of  engineering,  while  rational  mechanics 
developed,  without  regard  to  friction,  in  the  mathematical  field. 


CHAPTER    TEN 


LAZARE  CARNOTS  MECHANICS 


1 .  CARNOT  AND  THE  EXPERIMENTAL  CHARACTER  OF  MECHANICS. 

In  1783  Lazare  Carnot  (1753-1823)  published  an  Essai  sur  les 
machines  en  general.  He  later  extended  this  under  the  title  of  Principes 
generaux  de  Fequilibre  et  du  mouvement  (1803).  In  this  interval  La- 
grange  published  the  first  edition  of  his  Mecanique  analytique  (1788). 
But  Carnot's  ideas  varied  so  little  from,  the  Essai  to  the  Principes 
that  it  can  be  maintained  that  Lagrange  had  no  influence  on  Carnot. 
Further,  it  is  natural  to  think  of  Carnot  as  a  predecessor  of  Lagrange, 
in  spite  of  details  of  simple  chronology.1 

In  the  field  of  principles,  we  are  indebted  to  Carnot  because  he 
was  the  first  to  assert  the  experimental  character  of  mechanics — 
universally  accepted  now.  This  is  cpiite  in  contrast  with  the  ideas 
professed  by  Euler,  and  more  often,  by  d'Alembert.  The  declarations 
which  follow  are  taken  from  the  Principes  and  are  to  be  contrasted, 
in  particular,  with  the  introduction  to  d'Alembert's  treatise. 

"  The  Ancients  established  the  axiom  that  all  our  ideas  come 
from  our  senses  ;  and  this  great  truth  is,  today,  no  longer  a  subject 
of  controversy.  . .  .  [Here  Carnot  is  invoking  Locke's  Essay  on  Human 
Understanding.] 

"  However,  all  the  sciences  do  not  draw  on  the  same  experimental 
foundation.  Pure  mathematics  requires  less  than  all  the  others  ; 
next  come  the  physico -mathematical  sciences ;  then  the  physical 
sciences.  .  .  . 

"  Certainly  it  would  be  satisfactory  to  be  able  to  indicate  exactly 

1  CARNOT  himself  wrote,  in  the  preface  to  the  Principes,  "  Since  the  first  edition 
of  this  work  in  1783,  under  the  title  of  Essai  sur  les  machines,  there  have  appeared,  in 
all  branches  of  mechanics,  works  of  such  beauty  and  of  such  scope  that  there  hardly 
remains  room  for  some  remembrance  of  mine.  However,  as  it  contained  some  ideas 
that  were  new  at  the  time  it  appeared,  and  as  it  is  always  valuable  to  contemplate  the 
fundamental  truths  of  science  from  the  various  points  of  view  that  can  be  chosen,  a 
new  edition  has  been  asked  of  me. ..." 


324  THE   PRINCIPLES   OF  CLASSICAL   MECHANICS 

the  point  at  which,  each  science  ceased  to  be  experimental  and  became 
entirely  rational  [read,  in  order  to  develop  rationally,  starting  from 
principles  obtained  from  experiment]  ;  that  is,  to  be  able  to  reduce 
to  the  smallest  number  the  truths  that  it  is  necessary  to  infer  from 
experiment  and  which,  once  established,  suffice  to  embrace  all  the 
ramifications  of  the  science,  being  combined  by  reason  alone.  But 
this  seems  to  be  very  difficult.  In  the  desire  to  penetrate  more  deeply 
by  reason  alone,  it  is  tempting  to  give  obscure  definitions,  vague  and 
inaccurate  demonstrations.  It  is  less  inconvenient  to  take  more  in 
formation  from  experiment  than  would  strictly  be  necessary.  The 
development  may  seem  less  elegant.  But  it  will  be  more  complete 
and  more  secure.  .  .  . 

"  It  is  therefore  from  observation  that  men  derived  the  first 
concepts  of  mechanics.  However,  the  fundamental  laws  of  equilibrium 
and  motion  which  serve  as  its  foundation  offer  themselves  so  naturally 
to  reason  on  the  one  hand,  and  on  the  other,  show  themselves  so  clearly 
in  the  most  common  facts,  that  it  is  difficult  to  say  whether  it  is  from 
the  one  rather  than  from  the  other  that  we  derive  our  perfect  conviction 
of  these  laws ;  and  whether  this  conviction  would  exist  without  the  con 
currence  of  these  laws  with  the  first.  These  facts  seem  too  familiar  for  us 
to  be  able  to  know  at  what  point,  without  them,  reason  alone  would 
be  able  to  establish  definitions.  And,  on  the  other  hand,  if  reason 
is  unable  to  connect  these  facts  by  analogy,  they  appear  too  isolated 
for  us  to  be  able  to  weld  them  into  principles.  "  l 

2.  THE  CONCEPTS  AND  POSTULATES  OF  CARNOT'S  MECHANICS. 

Carnot  had  certainly  studied  Euler  and  d'Alembert,  and  thus 
knew  of  the  theory  of  forces  and  also  of  that  of  motions  (in  the  purely 
kinematic  sense).  He  reproaches  the  first  for  "  being  founded  on  a 
metaphysical  and  obscure  notion  of  forces.  "  If,  on  the  contrary, 
the  word  force  is  understood  to  be  the  momentum  impressed  on  a 
system,  the  first  theory  reduces  to  the  second  and  requires  an  appeal 
to  experiment. 

At  least  in  principle,  Carnot  adopts  the  second  attitude  and  seeks 
to  reduce  mechanics  to  the  study  of  the  communication  of  motion.  He 
applies  the  laws  of  mechanics  to  the  reasoned  observation  of  problems  of 
impact.  He  then  reduces  the  action  of  a  continuous  force  to  that 
of  a  series  of  infinitely  small  impacts. 

"  Weight  and  all  forces  of  the  same  kind  act  in  imperceptible 
degrees  and  produce  no  sudden  changes.  However,  it  seems  rather 

1  Principes  generaux  de  Vequilibre  et  du  Tnouvement,  p.  2. 


LAZAKE    CARNOT  325 

natural  to  consider  them  as  dealing  infinitely  small  blows,  at  infinitely 
short  intervals,  to  the  bodies  which  they  actuate.  " 

Thus  the  fundamental  law  of  Carnot's  mechanics  is  written,  apart 
from  notation,  in  the  form 

Fdt  =  d(mv). 

But  Carnot  accompanied  this  fundamental  law  with  the  following 
commentary. 

u  At  first  I  shall  repeat  that  the  question  here  is  not  that  of  the 
original  causes  which  create  motion  in  bodies,  but  only  that  of  the  motion 
already  produced  and  inherent  in  each  of  them.  The  quantity  of 
motion  already  produced  in  a  body  is  called  its  force  or  its  power. 
Thus  the  forces  which  are  considered  in  mechanics  are  not  metaphysical 
or  abstract  entities.  Each  of  them  resides  in  a  determinate  mass. 
The  force  is  the  product  of  this  mass  and  the  velocity  which  the  body  takes 
if  it  is  not  obstructed  by  the  motions  of  other  bodies  which  are  incompatible 
with  its  own.  Such  incompatibility  makes  some  bodies  lose  a  part 
of  their  quantity  of  motion  ;  it  makes  others  add  to  it,  and  creates 
it  in  those  which  had  none.  Each  body  assumes  a  kind  of  combined 
velocity,  in  between  the  one  which  it  must  have  already  had  and  those 
which  are  newly  impressed  on  all  its  parts.  Now  it  is  this  compound 
velocity  that  it  is  necessary  to  determine,  at  each  instant  and  for 
each  point  of  the  system,  when  the  shapes  of  the  different  parts  which 
compose  it,  their  masses  and  the  velocities  which  they  are  supposed 
to  have  received  previously — whether  by  earlier  impacts  or  by  external 
agencies  of  any  kind — are  known.  Thus,  in  a  word,  we  do  not  seek 
the  laws  of  motion  in  general,  but  rather  the  laws  of  the  communication 
of  motion  between  the  different  material  parts  of  a  single  system.  "  l 

In  fact,  Carnot  did  not  rigorously  dispense  with  the  concept  of 
force.  It  may  even  be  said  that  he  multiplied  the  names  for  it,  as 
we  shall  see.  Moreover,  this  conforms  with  his  general  attitude — 
his  mechanics  did  not  depend  on  a  closed  set  of  axioms. 

Carnot  variously  called  the  product  of  a  body's  mass  and  the 
accelerating  force  [read  "  acceleration"]  its  motive  force,  force  of  pressure 
or  dead  force.  Thus  gravity  or  heaviness  is  an  accelerating  force  and 
weight,  a  motive  force. 

By  moving  force  Carnot  understood  "  the  motive  force  applied  to 
a  machine  in  order  to  overcome  the  resistances,  or  to  produce  any 
motion  at  all.  "  If  the  living  force  is  expressed  by  the  product  Trav2, 
the  latent  living  force  is  expressed  by  the  product  PH  of  a  weight  and 

1  Principes  gGneraux  de  Fequilibre  et  du  mouvement,  p.  47. 


326  THE   PRINCIPLES   OF  CLASSICAL   MECHANICS 

a  height.     The  elementary  work  of  a  force  is  called,  hy  Camot,  the 
moment  of  activity  achieved  by  a  motive  force. 

As  for  the  moment  of  absolute  activity  of  a  moving  body,  this  can  be 
expressed  in  modern  language  by  the  product 

mv  (v  +  dv) 

where  v  +  dv  is  the  velocity  of  the  body  at  the  time  t  +  dt  (if  the  motion 
is  continuous) .    In  impact,  the  same  moment  of  activity  would  be  written 

TTii;  (v  -f-  Av) 

where  Av  is   a   finite  increment. 

Carnot  next  introduces  the  force  of  inertia  by  means  of  the  following 
definition —  "  The  resistance  offered  by  a  body  to  a  change  of  state  " 
or  the  "  reactions  opposed  to  a  system  of  bodies  which  make  it  pass 
from  rest  to  motion.  "  For  example,  in  an  impact  (the  external 
actions  being  supposed  negligable)  the  force  of  inertia  of  a  body  of  mass 
m  whose  velocity  changes  from  tT0  to  t^  would  be,  in  Carnot's  sense, 
m  (£TQ  —  JTJ.  Here  the  force  of  inertia  coincides  with  the  quantity 
of  motion  lost.  But,  in  general,  the  quantity  of  motion  lost  is  the  "re 
sultant  of  the  quantity  of  motion  produced  by  the  motive  force  and 
the  quantity  of  motion  produced  by  the  force  of  inertia.  "  Finally, 
Carnot  understands  the  force  exerted  on  a  body  of  the  system  to  be  the 
resultant  of  the  motive  force  and  the  force  of  inertia. 

In  passing,  we  note  a  curious  discussion  on  this  subject.  In  his 
Sixty-Sixth  Letter  to  a  German  Princess,  Euler  had  criticised  the  ex 
pression  **  force  of  inertia  "  as  uniting  the  concept  of  force 
(capable  of  changing  the  state  of  a  body)  and  the  word  inertia  (express 
ing  the  property  of  a  body  that  tends  to  preserve  it  in  its  state). 

Carnot  objected  that  "  the  inertia  is  merely  a  property  which 
may  not  be  introduced  in  the  calculations,  while  the  force  of  inertia 
is  a  real  measurable  property  ;  it  is  the  quantity  of  motion,  which 
this  body  imparts  to  any  other  body,  that  displaces  it  from  its  state.  "  * 

Carnot  assumed  the  following  postulates  as  a  foundation  for  his 
mechanics. 

1)  The  principle  of  inertia. 

2)  A    system   in    equilibrium    remains    in    equilibrium    under    the 
application   of  forces   which   are   in   equilibrium   among   themselves. 

3)  In  a  system  of  forces  in  equilibrium,  each  force  is  equal  and 
opposed  to  the  geometric  sum  of  all  the  others. 

4)  "  The  quantities  of  motion  of  motive  forces  which,  in  a  system 
of  bodies,  destroy  each  other  at  all  times,  can  always  be  decomposed 

1  Principes  g$neraux  de  requilibre  et  du  mouvement,  p.  73. 


LAZARE    CARNOT  327 

into  other  forces  which  are,  taken  in  pairs,  equal  and  directly  opposed 
along  the  direction  of  the  straight  line  which  connects  the  two  bodies 
to  which  they  belong.  And,  in  each  of  these  bodies,  each  force  can 
be  regarded  as  nullified  by  the  action  of  the  other.  " 

5)  The  action  of  one  body  on  another  by  impact,  traction  or  pressure, 
only  depends  on  the  relative  velocity  of  the  bodies. 

6)  "  The  quantities  of  motion  or  the  dead  forces  which  the  bodies 
impress  on  each  other  through  threads  or  rods  are  directed  along  these 
threads  or  rods  ;  and  those  which  they  impress  on  each  other  by  impact 
or  pressure  are  directed  along  the  perpendicular  erected  at  their  common 
surface  at  the  point  of  contact.  " 

7)  Hypotheses  expressing  the  laws  of  inelastic,  elastic  and  partially- 
elastic  impact. 

Given  these  definitions,  Carnot  introduced  the  concept  of  geo 
metrical  motion  into  mechanics  in  the  following  way. 

"  Every  motion  which  is  imparted  to  a  system  of  bodies  and  which 
does  not  alter  the  intensity  of  the  action  which  they  exert  or  could 
exert  on  each  other  when  any  other  motions  whatever  are  imparted 
to  them,  will  be  called  a  geometrical  motion.  Then  the  velocity  which 
each  body  assumes  will  be  called  its  geometrical  velocity.  "  l 

Carnot  has  the  following  comment  to  make  about  this  concept. 

"  This  denomination  of  geometrical  motion  is  based  on  the  fact 
that  the  motions  concerned  have  no  effect  on  the  action  which  can 
be  exerted  between  the  bodies  of  the  system,  and  that  they  are  inde 
pendent  of  the  rules  of  dynamics.  .  .  ,  They  only  depend  on  the  con 
ditions  of  constraint  between  the  parts  of  the  system  and,  consequently, 
can  be  determined  by  geometry  alone. 

46  The  theory  of  geometrical  motions  is,  in  a  sense,  a  science  inter 
mediate  between  geometry  and  mechanics.  It  is  the  theory  of  the 
motions  that  a  system  of  bodies  can  assume  without  the  bodies  hinder 
ing  each  other,  or  exerting  any  action  or  reaction  on  each  other.  "  2 

In  modern  language,  Carnot's  geometrical  motions  are  virtual  dis 
placements  (finite  or  infinitely  small)  compatible  with  the  constraints 
between  the  bodies  of  the  system. 

3.  CARNOT'S  THEOREM. 

In  the  second  part  of  his  Principes  fondamentaux  Carnot  studied 
the  motion  of  systems,  taking  as  his  basis  the  problems  of  impact 
between  "  hard  bodies " — that  is,  bodies  devoid  of  elasticity. 

1  Principes  generaux  de  requilibre  et  du  mouvement,  p.  108. 
2  Ibid.,  p.  106. 


328  THE   PRINCIPLES  OF  CLASSICAL   MECHANICS 

Carnot  first  shows  that  "  if  a  system  of  hard  bodies  suffers  an 
impact  or  any  instantaneous  action,  either  directly  or  by  means  of 
some  mechanism  without  elasticity,  the  motion  taken  by  the  system 
is  necessarily  a  geometrical  one.  " 

Indeed,  if  the  bodies  contiguous  with  the  system  by  which  the 
action  is  propagated  are  considered  in  pairs,  after  the  impact  they 
have  no  relative  velocity  in  the  line  of  their  reciprocal  action.  Their 
real  motions  after  the  impact  cannot  therefore  produce  any  action 
between  them.  It  follows  that  the  motion  of  the  system  after  the 
impact  is  necessarily  a  geometrical  one.  Moreover,  it  is  easy  to  see 
that  every  geometrical  motion  which  is  imparted  to  any  system  is 
received  by  the  system  without  alteration. 

Turning  to  the  consideration  of  a  system  of  hard  bodies  which 
sustains  an  impact,  Carnot  decomposes  (after  the  manner  of  d'Alembert) 
the  motion  of  the  system  before  the  impact  into  two  others.  The 
first  of  these  is  that  which  remains  after  the  impact  and  the  second 
is,  consequently,  necessarily  destroyed  by  the  impact. 

If  only  the  first  motion  is  imparted  to  the  system,  it  will  necess 
arily  be  received  without  alteration. 

Under  the  influence  of  the  second  motion,  also  considered  in  isolation, 
the  system  remains  in  equilibrium. 

Carnot  writes,  "  This  is  what  constitutes  d'Alembert's  famous 
principle.  But  it  must  be  recalled  that  it  is  only  applicable  to  perfectly 
hard  bodies  and  to  mechanisms  without  elasticity — this,  I  think, 
has  not  been  observed  explicitly  before.  If  the  bodies  were  elastic, 
the  motion  before  the  impact  would  decompose  into  two  in  the  same 
way  as  for  hard  bodies.  One  of  these  motions  would  be  the  motion 
that  remains  after  the  impact  and  the  other  would  be  destroyed.  But 
the  independence  of  these  motions  would  not  subsist ;  for  if  the  first 
alone  were  suppressed,  there  would  not  be  equilibrium.  This  inde 
pendence  of  the  two  motions  is  based  on  the  fact  that  the  motion 
after  the  impact  is  geometrical ;  that  is,  it  does  not  tend  to  increase 
or  decrease  the  intensity  of  the  impact,  and  it  is  only  such  because 
the  bodies,  being  hard,  etc.  ..." 

Let  U  denote  the  velocity  lost  by  a  particle  M  during  the  impact 
and  let  V  be  its  velocity  after  the  impact. 

By  induction,  starting  from  TorricellVs  principle,  Carnot  states 
the  law 

(1)  $MUVcos(vTU)  =  Q. 

Here  indeed,  Carnot  makes  appeal  to  continuous  motions  by  starting 
from  the  axiom  that  "  when  the  centre  of  gravity  is  lowest,  the  system 


IAZARE    CARNOT  329 

is  in  equilibrium.  "  If  p  is  the  accelerating  force,  Carnot  writes  the 
condition  for  the  equilibrium  of  a  system,  in  continuous  motion  under 
the  influence  of  the  forces  p,  in  the  form 

SpMFcos  (pTV)  =  0. 

From  this  he  deduces  the  law  (1)  by  applying  this  principle  to  percussions. 

Carnot  verifies  the  law  (1)  for  the  particular  impact  of  two  hard 
bodies,  using  an  analysis  that  is,  this  time,  direct.  He  then  extends 
the  law  to  the  impact  of  any  number  of  hard  bodies. 

From  these  results,  Carnot  easily  deduced  the  following  theorem, 
with  which  his  name  is  still  associated. 

"  In  the  impact  of  hard  bodies,  the  sum  of  the  living  forces  before 
the  impact  is  always  equal  to  the  sum  of  the  living  forces  after  the  impact 
together  with  the  sum  of  the  living  forces  that  each  of  these  bodies  would 
have  if  it  moved  freely  with  only  the  velocity  which  it  lost  in  the  impact.  "  * 

Indeed,  it  was  sufficient  for  him  to  write 

SMTF2  =  SMF2  +  SM U2  +  2  $MVU  cos  (VTU] 

where   W  is  the  velocity  before  the  impact  and  law  (1)  is  applied. 

Using  d'Alembert's  procedures  throughout,  Carnot  treated  problems 
of  elastic  impact  as  corollaries  of  problems  of  impact  between  "  hard  " 
bodies.  The  elasticity  doubles  the  momentum  lost  without  changing 
its  direction.  Thus,  to  Carnot,  the  conservation  of  living  forces  in 
the  impact  of  perfectly  elastic  bodies  is  justified  by  his  theorem  on 
the  impact  of  hard  bodies. 

From  the  general  equation  (1)  Carnot  also  deduced  the  remarkable 
result  that  the  sum  of  the  living  forces  due  to  the  velocities  lost  is 
a  minimum  in  the  impact  of  a  system  of  hard  bodies. 

"  Among  the  motions  to  which  a  system  of  perfectly  hard  bodies  is 
susceptible,  when  the  bodies  act  on  each  other  by  a  direct  impact  or  by 
any  mechanism  without  elasticity,  so  that  there  results  a  sudden  change 
in  the  state  of  the  system,  the  one  that  actually  remains  after  the  action 
is  the  geometrical  motion  which  is  such  that  the  sum  of  the  products  of 
each  of  the  masses  by  the  square  of  the  velocity  that  it  loses  is  a  minimum ; 
that  is,  less  than  the  sum  of  the  products  of  the  masses  and  the  square 
of  the  velocity  that  it  would  have  lost  if  the  system  had  acquired  any  other 
geometrical  motion.  " 

Carnot  himself  remarked  that  this  result  was  directly  connected 
with  Maupertuis9  application  of  the  principle  of  least  action  to  the 
impact  of  bodies. 

1  Principes  genfraux  de  Fequilibre  et  du  mouvement^  p.  145. 


330  THE   PRINCIPLES   OF  CLASSICAL   MECHANICS 

In  this  connection,  Carnot  emerges  as  an  opponent  of  the  doctrine 
of  final  causes.  Indeed,  he  declares  that  his  demonstration  of  this 
minimum  law  "  is  more  general  [than  that  of  Maupertuis]  because 
it  includes  bodies  which  have  various  degrees  of  elasticity.  But  it 
also  demonstrates  how  insecure  are  those  which  are  based  on  final 
causes,  since  it  shows  that  the  principle  is  not  general,  but  restricted 
to  systems  of  bodies  which  have  the  same  degree  of  elasticity.  " 

Without  carrying  this  analysis  of  Carnot's  mechanics  further, 
we  shall  indicate  how  he  passed  from  the  study  of  these  problems 
of  impact  to  problems  in  which  continuous  forces  intervene. 

"  When  a  system  of  hard  bodies,  free  or  acted  upon  by  any  mechanism 
without  elasticity,  and  actuated  by  any  moving  forces,  changes  its  motion 
by  imperceptible  degrees  then  if,  it  any  instant  of  the  motion,  each  one 
of  the  particles  is  catted  m ;  its  velocity  V  ;  its  motive  force  P  x  ;  the  velocity 
that  it  would  take  if  the  actual  motion  were  suddenly  suppressed  and 
replaced  by  another  geometrical  one,  u  ;  the  element  of  time,  dt ;  then 
there  will  obtain 

Smud  [V  cos  (u^V)]  —  SmuPdt  cos  (u^P]  =  0. " 

This  theorem  is  deduced  from  the  general  formula  (1)  by  observing 
that 

Pdt  cos  (£TP)  —  d[V  cos  (uTP)] 

is  the  projection,  on  the  direction  of  u,  of  the  velocity  lost  by  the  mass 
m,  due  to  the  action  of  the  other  elements  of  the  system. 

Carnot  also  develops  some  very  interesting  considerations  on 
the  work  of  the  internal  forces  in  animal  systems. 

"  An  animal,  like  the  inanimate  bodies,  is  subject  to  the  law  of 
inertia.  That  is,  the  general  system  of  parts  which  compose  it  cannot 
produce  by  itself  any  progressive  motion  in  any  direction.  ...  In 
the  whole  system  of  the  animal,  the  principle  of  the  equality  of  the 
action  and  the  reaction  is  applicable,  as  in  inert  matter.  So  that  it 
is  only  by  the  friction  of  its  feet  on  the  ground  that  it  can  carry  itself 
forward,  thereby  impressing  on  the  earth  on  which  it  walks  a  quantity 
of  motion  equal  and  opposite  to  that  which  it  assumes,  but  which 
is  imperceptible  to  us. 

"  It  therefore  seems,  as  far  as  its  physique  is  concerned,  that  the 
animal  may  be  considered  as  an  assembly  of  particles  separated  by 
springs  which  are  more  or  less  compressed  and  which,  by  this  fact, 

1  Here  it  is  necessary  to  read  "  accelerating  force.  " 


LAZARE    CARNOT  331 

store  a  certain  quantity  of  living  forces  ;  and  that  these  springs,  by 
extending,  may  be  considered  to  convert  this  latent  living  force  into 
real  living  force.  .  .  . 

"  When  a  similar  agency  imparts  living  force  to  its  own  mass,  al 
though  the  quantity  of  motion  which  results  in  any  direction  may  be 
zero,  the  living  force  is  not  zero.  And  if  this  agency  is  applied  to 
a  machine,  its  acquired  living  force  will  be,  by  means  of  this  machine, 
transmitted  to  the  resisting  forces  without  loss — always  with  the 
reservation  that  there  should  be  no  impacts ;  for  what  will  be  consumed 
will  be  wholly  absorbed  and  will  be  precisely  what  we  call  the  effect 
produced.  "  1 

The  general  conclusion  of  Carnot's  mechanics  is  the  following 
one. 

"  For  any  system  of  bodies,  animated  by  any  motive  forces,  in 
which  several  external  agents  such  as  men  or  animals — either  by 
themselves  or  by  machines — are  used  to  move  the  system  in  different 
ways,  whatever  may  be  the  change  produced  in  the  system,  the 
moment  of  activity  consumed  by  the  external  powers  in  any  time 
will  always  be  equal  to  half  the  amount  by  which  the  sum  of  the  living 
forces  in  the  system  of  bodies  to  which  they  are  applied  will  be  increased 
during  this  time,  less  half  the  amount  by  which  this  same  sum  of  living 
forces  would  be  increased  if  each  of  the  bodies  had  moved  freely  on 
the  curve  which  it  described — supposing  that  it  had  experienced  the 
same  motive  force,  at  each  point  of  this  curve,  as  that  which  it  actually 
experienced  ;  and  provided  always  that  the  motion  changes  by  imper 
ceptible  degrees,  so  that,  if  machines  with  springs  are  used,  these 
springs  are  left  in  the  same  state  of  tension  as  at  the  beginning.  " 

Certainly  Carnot's  language  did  not  approach  the  clarity  of  the 
great  authors  of  the  Century,  But  the  foundation  of  his  work  is 
of  an  undisputed  originality,  at  once  physical  and  philosophical. 
In  fact,  Lazare  Carnot  was  to  inspire  Laplace,  Barre  de  Saint- Venant 
and  probably  Coriolis  as  well. 

1  Principes  ggneraux  de  Vequilibre  et  du  mouvement,  p.  246. 


CHAPTER   ELEVEN 


THE  "  MECANIQUE  ANALYTIQUE  "  OF  LAGRANGE 


1.    THE    CONTENT   AND    PURPOSE    OF    LAGRANGE'S    "  MECANIQUE 
ANALYTIQUE.  " 

We  now  come  to  a  piece  of  work  which,  united  and  crowned  all  the 
efforts  which  were  made  in  the  XVIIIth  Century  to  develop  a  rationally 
organised  mechanics. 

Coming  from  a  Touraine  family,  Louis  de  Lagrange  (1736-1813) 
started  his  career  at  Turin,  where  he  had  been  born.  After  having 
come  under  the  influence  of  Euler  at  the  Academy  of  Berlin,  he  finally 
went  to  Paris  in  1787  where,  in  particular,  he  inaugurated  the  teaching 
of  analysis  at  the  £cole  polytechnique.  Thus,  by  his  descent  and  for 
an  important  part  of  his  scientific  career,  Lagrange  belonged  to  France. 

The  first  edition  of  the  Mecanique  analytique  appeared  in  1788,1  In 
it  Lagrange  accomplished  the  project,  which  had  been  conceived 
and  partially  executed  by  Euler,  of  a  single  treatise  of  rational  science 
(analytice  exposita)  covering  all  branches  of  mechanics,  statics  and 
hydrostatics,  dynamics  and  hydrodynamics. 

Lagrange's  reading  covered  everything.  Apart  from  the  works  of 
his  contemporaries,  he  had  studied,  with  a  remarkable  objectivity, 
those  of  all  the  ancient  and  modern  writers  that  were  known  in  his 
time.  This  is  witnessed  by  the  historical  references  with  which  he 
enriched  his  treatise. 

Lagrange  eliminated  the  contradictions  and  the  inarticulateness 
which  abounded  in  the  work  of  his  predecessors.  He  adopted  the 
concepts  and  the  postulates  of  the  great  creators  of  the  previous  century 
(Galileo,  Huyghens,  Newton).  He  surpassed  Euler  and  d'Alembert. 
And  he  became  preoccupied  with  the  organisation  of  mechanics,  the 

1  The  last  edition  to  be  published  in  LAGRANGE'S  lifetime  appeared  in  1811.  In  the 
present  book  we  have  made  use  of  the  edition  of  1853-1855,  which  was  amended  by 
Joseph  BERTRAND  and  used  certain  manuscripts  which  had  not  been  published  during 
LAGRANGE'S  life. 


LAGRANGE  333 

foundation  of  its  principles,  the  perfection  of  its  mathematical  language 
and  the  isolation  of  a  general  analytical  method  for  solving  its  problems. 
His  clarity  of  mind,  his  mathematical  insight,  served  him  so  well  that 
he  arrived  at  an  almost  perfect  codification  of  mechanics  in  the  class 
ical  field.  In  a  detailed  way,  Lagrange  made  the  following  statement 
of  his  aims  in  an  Avertissement. 

"  To  reduce  the  theory  of  mechanics,  and  the  art  of  solving  the 
associated  problems,  to  general  formulae,  whose  simple  development 
provides  all  the  equations  necessary  for  the  solution  of  each  problem. 

"  To  unite,  and  present  from  one  point  of  view,  the  different  prin 
ciples  which  have,  so  far,  been  found  to  assist  in  the  solution  of  problems 
in  mechanics  ;  by  showing  their  mutual  dependence  and  making  a 
judgement  of  their  validity  and  scope  possible.  " 

As  for  the  purely  mathematical  point  of  view  which  was  Lagrange 's 
principal  interest,  he  made  the  following  declaration. 

"  No  diagrams  will  be  found  in  this  work.  The  methods  that  I 
explain  in  it  require  neither  constructions  nor  geometrical  or  mechanical 
arguments,  but  only  the  algebraic  operations  inherent  to  a  regular 
and  uniform  process.  Those  who  love  Analysis  will,  with  joy,  see 
mechanics  become  a  new  branch  of  it  and  will  be  grateful  to  me  for 
thus  having  extended  its  field.  " 

2.  LAGRANGE'S  STATICS. 

In  the  historical  part  of  his  work  Lagrange  makes  special  mention 
of  Archimedes,  Stevin,  Galileo  and  Huyghens.  In  his  view,  the  equi 
librium  of  a  straight  and  horizontal  lever  whose  ends  are  loaded  with 
equal  weights  and  whose  point  of  support  is  at  the  centre  is  "  a  truth 
that  is  evident  on  its  own. "  On  the  other  hand,  the  principle  of  the 
superposition  of  equilibria,  as  fruitful  as  the  principle  of  the  super 
position  of  figures  in  geometry,  is  essential  for  a  treatment  of  the 
angular  lever.  This  leads  to  the  principle  of  moments,  in  which  connec 
tion  Lagrange  cites  Guido  Ubaldo. 

Lagrange  refers  to  Stevin  and  to  Galileo's  mechanics  in  connection 
with  the  inclined  plane.  In  the  matter  of  the  decomposition  of  a 
force  into  its  components,  he  places  Roberval  before  Stevin. 

To  Lagrange,  Descartes'  principle  and  that  of  Torricelli  were  put 
forward  without  proof  by  their  authors. 

Lagrange  mentions  Aristotle,  Archimedes,  Nicomedes  and,  among 
the  moderns,  Descartes,  Wallis  and  Roberval,  as  having  used  the 
composition  of  motions.  It  was  Galileo  who  had  made  first  use  of 
this  concept  in  dynamics,  in  connection  with  the  motion  of  projectiles. 


334  THE   PRINCIPLES   OF  CLASSICAL   MECHANICS 

But,  with  good  reason,  Lagrange  attributes  the  composition  of  forces, 
in  the  proper  sense  of  the  term,  to  Newton,  Varignon  and  Lamy.  An 
immediate  connection,  which  Varignon  saw  and  demonstrated  by  the 
theory  of  moments,  exists  between  the  principle  of  the  lever  and  that 
of  the  composition  of  forces. 

Lagrange  gives  the  following  opinion  on  the  justification  of  the 
rule  of  the  parallelogram  which  had  been  given  by  Daniel  Bernoulli. 
"  By  separating,  in  this  way,  the  principle  of  the  composition  of  forces 
from  the  principle  of  the  composition  of  motion,  the  principal  advantages 
of  clarity  and  simplicity  were  lost,  and  the  principle  was  reduced  to 
being  merely  the  result  of  geometrical  constructions  and  analysis.  " 

Lagrange  then  comes  to  the  principle  of  virtual  work,  which  he  states 
in  the  following  way. 

"  Powers  are  in  equilibrium  when  they  are  inversely  proportional 
to  their  virtual  velocities  taken  in  their  own  directions." 

Lagrange  mentions  Guido  Ubaldo  as  having  been  concerned  in 
the  formation  of  this  principle.  He  refers  to  the  concept  of  momento 
as  used  in  Galileo's  statics,  recalls  the  part  played  by  Descartes  and 
Torricelli  and  honours  Jean  Bernoulli  for  having  been  the  first  to 
formulate  the  principle  in  all  its  generality. 

The  justification  of  the  principle  of  virtual  work  occupies  a  great 
deal  of  Lagrange's  attention.1 

"  As  for  the  nature  of  the  principle  of  virtual  velocities,  it  must 
be  agreed  that  it  is  not  sufficiently  clear  in  itself  to  be  formed  into  a 
first  principle.  But  it  can  be  regarded  as  the  general  expression  of 
the  laws  of  equilibrium,  deduced  from  two  principles  [of  the  lever 
and  of  the  composition  of  forces].  Further,  in  the  demonstrations 
of  this  principle  which  have  been  given,  it  has  always  been  made  to 
depend  on  these  by  means  which  are  more  or  less  direct.  But  there 
is  another  general  principle  in  statics  which  is  independent  of  the 
principle  of  the  lever  and  the  principle  of  the  composition  of  forces 
which,  although  it  is  customarily  related  to  the  others  in  mechanics, 
appears  to  be  the  natural  foundation  of  the  principle  of  virtual  velocities 
— it  can  be  called  the  principle  of  pulleys. 

"  If  several  pulleys  are  mounted  together  on  a  single  frame  this 
assembly  is  called  a  polispaste  or  pulley-block.  The  combination  of 
two  pulley-blocks — one  fixed  and  the  other  moveable — which  is  wound 
with  a  single  string,  one  end  of  which  is  permanently  attached  and  the 
other,  acted  upon  by  a  power,  forms  a  machine  in  which  the  power 
is  to  the  weight  carried  by  the  moveable  pulley-block  as  unity  is  to 
the  number  of  strands  which  converge  on  this  pulley-block  ;  this, 
1  Mecanique  analytigue,  Vol.  I,  p.  21. 


LAGRANGE  335 

if  the  strands  are  all  supposed  to  be  parallel  and  the  friction  and  the 
stiffness   of  the  strings  is  neglected. 

"  By  increasing  the  numbers  of  fixed  and  moveable  pulley- 
blocks,  and  winding  them  all  with  the  same  string  by  means  of 
various  fixed  and  reversing  pulleys,  the  same  power,  when  it  is 
applied  to  the  moveable  end,  will  be  able  to  support  as  many 
weights  as  there  are  moveable  pulley-blocks.  Then,  each  weight  will 
be  to  the  power  as  the  number  of  strands  of  the  pulley-block  support 
ing  it  is  to  unity. 

"  For  greater  simplicity,  make  the  last  strand  pass  over  a  fixed 
pulley  and  let  it  support  a  weight  instead  of  the  power.  We  shall 
assume  this  weight  to  be  unity.  Also  imagine  that  the  different 
moveable  pulley-blocks,  instead  of  supporting  weights,  are  attached 
to  bodies — regarded  as  points — and  arranged  among  each  other  so 
that  they  form  any  given  system.  In  this  way,  by  means  of  the  string 
which  is  wound  round  all  the  pulley-blocks,  the  same  weight  will 
produce  various  powers,  which  act  on  the  different  parts  of  the  system 
in  the  direction  of  the  strings  which  converge  on  the  pulley-blocks 
attached  to  these  points.  The  powers  will  be  to  the  weight  as  the 
number  of  strands  is  to  unity.  So  that  the  powers  themselves  will 
be  represented  by  the  number  of  strands  which  come  together  and, 
by  their  tension,  produce  them. 

"  Now  it  is  clear  that  in  order  that  the  system  drawn  by  these 
different  powers  may  remain  in  equilibrium,  it  is  necessary  that  the 
weight  should  be  unable  to  descent  by  any  infinitely  small  displacement 
of  the  points  of  the  system.  For  since  the  weight  always  tends  to  descend, 
if  there  is  any  infinitely  small  displacement  of  the  system  which  allows 
it  to  descend,  it  will  necessarily  do  so  and  will  produce  this  displa 
cement  of  the  system. 

"  Denote  the  infinitely  small  distances  which  this  displacement 
would  make  the  different  points  of  the  system  travel  by  a,  /?,  y,  .  .  . 
in  the  direction  of  the  power  which  pulls  them.  Also  denote  the 
number  of  strands  of  the  pulley-blocks  applied  at  these  points,  to 
produce  these  powers,  by  P,  $,  JZ,  .  .  .  It  can  be  seen  that  the 
distances  a,  /?,  y, .  .  .  will  also  be  those  by  which  the  moveable 
pulley-blocks  approach  the  associated  fixed  pulley-blocks.  Further, 
it  can  be  seen  that  these  movements  will  decrease  the  length  of  the  string 
which  is  wound  round  all  the  pulley-blocks  by  the  quantities  Pa, 
()/9,  fty,  ...  So  that,  because  of  the  fixed  length  of  the  string,  the 
weight  will  descend  throughout  the  distance 

P«  +  QB  +  Ry  +  . . . 


336  THE   PRINCIPLES   OF  CLASSICAL   MECHANICS 

"  Therefore,  in  order  that  the  powers  represented  by  the  numbers 
P,  $,  JR,  .  .  .  may  be  in  equilibrium,  it  will  be  necessary  that  the  equation 

Pa  +  Q$  +  Ry  +  . . .  =  0 

should  obtain.     This  is  the  analytic  expression  of  the  general  principle 
of  virtual  velocities.  " 

We  remark  here,  with  Jouguet, l  that  Lagrange's  demonstration 
is  based  on  physical  facts — on  certain  simple  properties  of  pulleys 
and  strings.  Lagrange  also  assumes  the  truth  of  the  principle  in  a 
very  particular  case,  which  reduces  to  the  hypothesis  of  Huyghens 
and  Torricelli. 

We  owe  to  Lagrange  the  elegant  method  called  that  of  multipliers. 
The  object  of  this  was  to  express,  in  a  general  way,  the  problems  of 
statics  by  means  of  mathematical  equations.2 

Lagrange  expressed  the  constraints  of  the  system  by  equations 
of  the  type 

L  =  0         M=  0         N  =  0  ... 

where  L,  M,  N  are  finite  functions  of  the  coordinates  of  the  points 
of  the  system. 

Differentiating  these  conditions,  Lagrange  writes 

dL  =  0         AM  =  0         dN  =  0  . . . 

(He  does  not  exclude  equations  of  constraint  between  differentials 
that  are  not  "  exact  differences  " — these  are  the  constraints  that  are 
now  called  non-holonomic.) 

Lagrange  declares,  "  These  equations  should  only  be  used  to  elim 
inate  a  similar  number  of  differentials  in  the  general  formula  of  equi 
librium,  after  which  the  coefficients  of  the  remaining  differentials 
all  become  equal  to  zero.  It  is  not  difficult  to  show,  by  the  theory 
of  the  elimination  of  linear  equations,  that  the  same  result  will  obtain 
if  the*  various  equations  of  condition 

dL  =  0,    dM  =  0,    dN  =  0,  . . . 

are  each  multiplied  by  an  indeterminate  coefficient  and  simply  added 
to  the  equation  concerned ;  if  then,  the  sum  of  all  the  terms  which 
are  multiplied  by  the  same  differential  are  equated  to  zero,  which  will 
give  as  many  particular  equations  as  there  are  differentials  ;  and  if, 
finally,  the  indeterminate  coefficients  by  which  the  equations  of  con- 

1  L.  M.,  Vol.  II,  p.  179. 

2  Mecanique  analytique.  Vol.  I,  p.  69  et  seq. 


LAGRANGE  337 

dition  have  been  multiplied  are  eliminated  from  the  last  set  of  equation.  " 
Whence  the  rule  stated  by  Lagrange  for   finding  the  conditions 
of  equilibrium  of  any  system  — 

"  The  sum  of  the  moments  [that  is,  apart  from  sign,  the  virtual 
works]  of  all  the  powers  which  are  in  equilibrium  will  be  taken,  and  the 
differential  functions  which  become  zero  because  of  the  conditions 
of  the  problem  will  be  added  to  it,  after  each  of  these  functions  has 
been  multiplied  by  an  indeterminate  coefficient  ;  then  the  whole  will 
be  equated  to  zero.  Thus  will  be  obtained  a  differential  equation 
which  will  be  treated  as  an  ordinary  equation  of  maximis  et  minimis. 
From  this  will  be  deduced  as  many  equations  as  there  are  variables. 
These  equations,  being  then  rid  of  the  indeterminate  coefficients  by 
elimination,  will  provide  all  the  conditions  necessary  for  equilibrium. 
"  The  differential  equation  concerned  will  therefore  be  of  the  form 

Pdp  +  Qdq  +  Rdr  +  ...  +  UL  +  pdM  +  vdN  +  .  .  .  =  0 

in  which  A,  /*,  v  are  the  indeterminate  quantities.     In  the  sequel  we 
shall  call  this  the  general  equation  of  the  equilibrium. 

"  Corresponding  to  each  coordinate  of  each  body  of  the  system, 
such  as  x,  this  equation  will  give  an  equation  of  the  form 


P3?   ,    n  39   ,    P  dr  _L  .    i  9L    t       BM    .       dN   L  n 

Pf-  +  Q^  +  R^~  +   ...   +  A-  --  h/*-a  --  h  V  —  +...=  0. 

dx  ox  dx  dx       ^  dx  ox 

Therefore  the  number  of  these  equations  will  be  equal  to  the  number 
of  all  the  coordinates  of  all  the  bodies.  We  shall  call  these  the  particular 
equations  of  the  equilibrium.  " 

It  only  remains  to  eliminate  the  multipliers  A,  u,  v.  Taking  account 
of  the  equations  of  constraint,  the  problem  of  the  determination  of 
the  coordinates  of  the  different  elements  of  the  system  is  thus  solved. 

Lagrange  did  not  confine  himself  to  this  abstract  analysis,  but 
gave  it  a  physical  interpretation.  The  terms  AdL,  /idM,  vdN  "  must 
be  regarded  as  representing  the  moments  [of  virtual  works]  of  certain 
forces  applied  to  a  system.  " 

Thus  dL  is  written  in  the  form 

dL  (*',  /,  *',  *",  /',  *"...)  =  dL'  +  *L"+  ... 

In  this  equation  (x,  y\  *'),  (#",  y",  2"),  etc.  .  .  .  represent  the  coor 
dinates  of  each  particle,  and  dl/,  dL'  \  etc.  .  .  .  only  depend  on  (x\y\  zr), 
(x",  y",  z"),  etc.  .  .  .  respectively.  Lagrange  then  verifies  that  the 
term  hdL  is  equivalent  to  the  effect  of  different  forces 


338  THE   PRINCIPLES  OF  CLASSICAL   MECHANICS 


applied,  respectively,  at  the  points  (#',  y',  z'),  (#",  y",  z"),  etc.  .  .  . 
and  normal  to  the  different  surfaces  defined  by  the  equation  dL  =  0. 
In  this  equation  the  variation  is  first  performed  with  respect  to  (#',  y',  2'), 
then  with  respect  to  (#",  y",  2"),  etc.  . .  . 

Lagrange  concludes,  "  It  follows  from  this  that  each  equation 
of  condition  is  equivalent  to  one  or  more  forces  applied  to  the  system 
in  given  directions.  So  that  the  state  of  equilibrium  of  the  system 
will  be  the  same  whether  the  consideration  of  forces  is  used,  or  whether 
the  equations  of  condition  themselves  are  used. 

"  Conversely,  these  forces  must  take  the  place  of  the  equations 
of  condition  resulting  from  the  nature  of  the  given  system,  so  that 
by  making  use  of  these  equations  it  will  be  possible  to  regard  the 
bodies  as  entirely  free  and  without  any  restraint.  And  from  this  is 
seen  the  metaphysical  reason  why  the  introduction  of  the  terms  Ad£  -f 
judM  +  ...  in  the  general  equation  of  equilibrium  ensures  that  this 
equation  can  then  be  treated  as  if  all  the  bodies  were  entirely  free.  .  .  . 

"  Strictly  speaking,  the  forces  in  equation  take  the  place  of  the 
resistances  that  the  bodies  would  suffer  because  of  their  mutual  con 
straint  or  because  of  obstacles  which,  by  the  nature  of  the  system, 
could  oppose  their  motion  ;  or  rather,  these  forces  are  merely  the  same 
forces  as  the  resit ances,  which  are  equal  and  directly  opposite  to  the 
pressures  exerted  by  the  bodies.  As  is  seen,  our  method  provides 
a  means  of  determining  these  forces  and  resistances.  .  .  .  " 

The  considerable  progress  achieved  by  Lagrange  in  the  analytical 
application  of  the  principle  of  virtual  work  is  very  evident. 

Lagrange  does  not  become  inordinately  eloquent  on  the  concept 
of  force  itself.  He  confines  himself  to  saying,  "  By  force  or  power 
is  understood,  in  general,  the  cause  which  imparts,  or  tends  to  impart, 
motion  to  the  bodies  to  which  it  is  supposed  to  be  applied  ;  further, 
it  is  by  the  quantity  of  motion  imparted,  or  which  may  be  imparted, 
that  the  force  must  be  represented.  In  the  state  of  equilibrium  the 
force  does  not  have  actual  effect ;  it  only  provides  a  tendency  to  motion. 
But  it  can  always  be  measured  by  the  effect  that  it  would  produce 
if  it  were  not  arrested.  " l 

1  M&anique  analytique^  Vol.  I,  p.  1. 


LAGRANGE  339 

3.  LAGRANGE  AND  THE  HISTORY  OF  DYNAMICS. 

In  Lagrange's  view,  dynamics  is  "  the  science  of  accelerating  or 
retarding  forces  and  the  varying  motions  which  they  must  produce. 
This  science  we  owe  entirely  to  the  moderns,  and  Galileo  is  the  one 
who  laid  its  first  foundations.  .  .  .  Huyghens,  who  seems  to  have  been 
destined  to  perfect  and  complete  most  of  Galileo's  discoveries,  sup 
plemented  the  theory  of  heavy  bodies  by  the  theories  of  the  motion 
of  pendulums  and  centrifugal  forces,  and  thus  prepared  the  way  for 
the  great  discovery  of  universal  gravitation.  Mechanics  became  a 
new  science  in  the  hands  of  Newton,  and  his  Principia,  which  appeared 
in  1687,  was  the  occasion  of  this  revolution.  " I  Thus,  neglecting 
all  the  vicissitudes  of  Aristotelian  mechanics  and  the  few  inspirations 
of  the  Schoolmen,  Lagrange  acknowledged  a  century  of  evolution  in  the 
subject  that  he  was  to  codify. 

Lagrange  ascribes  the  two  principles  of  the  force  of  inertia  (that 
is,  inertia)  and  the  composition  of  motions  to  Galileo.  He  analyses  the 
method  followed  by  Huyghens  in  his  work  on  the  centrifugal  force 
in  the  following  way. 

"  For  the  estimation  of  forces,  it  suffices  to  consider  the  motion 
produced  in  any  time,  finite  or  infinite,  provided  that  the  force 
may  be  regarded  as  constant  during  this  time.  Consequently,  whatever 
the  motion  of  the  body  and  the  law  of  acceleration  may  be,  since,  by 
the  properties  of  the  differential  calculus  the  action  of  every  accele 
rating  force  may  be  regarded  as  constant  during  an  infinitely  small 
time,  it  will  always  be  possible  to  find  the  value  of  the  force  which 
acts  on  the  body  at  each  instant.  This  is  done  by  comparing  the 
velocity  produced  in  this  instant  with  the  duration  of  the  same  instant ; 
or  by  comparing  the  distance  which  the  body  travels  with  the  square 
of  the  duration  of  the  same  instant.  It  is  not  necessary,  even,  that 
the  distance  should  be  actually  travelled  by  the  body,  it  is  sufficient 
that  it  may  be  supposed  to  have  been  travelled  by  a  compound  motion, 
since  the  effect  of  the  force  is  the  same  in  one  case  as  in  the  other.  "  2 

In  a  careful  analysis  of  the  use  of  mathematics,  Lagrange  remarks 
that  "  Newton  made  constant  use  of  the  geometric  method  as  simplified 
by  the  consideration  of  the  first  and  last  ratios.  "  Euler's  Mechanica 
(1736)  is,  to  Lagrange,  the  first  great  work  in  which  Analysis  was 
applied  to  the  science  of  motion.  As  for  MacLaurin's  Treatise  on 
Fluxions  (1742),  this  was  the  first  work  which  systematically  used 

1  Mfaanique  analytique,  Vol.  I,  p.  207. 

2  Ibid.,  p.  210. 


340  THE    PRINCIPLES  OF  CLASSICAL   MECHANICS 

the  rectangular  components  of  the  force  instead  of  their  tangential 
and  normal  components. 

Lagrange  then  comes  to  a  principle  which  allows  the  determination 
of  the  force  on  bodies  in  motion,  having  regard  to  their  mass  and  velocity. 
66  This  principle  consists  in  that,  in  order  to  impart  to  a  given  mass 
a  certain  velocity  in  some  direction,  whether  the  mass  be  at  rest  or 
in  motion,  the  necessary  force  is  proportional  to  the  product  of  the  mass 
and  the  velocity  and  its  direction  is  the  same  as  that  of  the  velocity.  "  l 

Here  Lagrange  cites  Descartes  as  having  first  realised  the  existence 
of  this  principle,  but  as  having  deduced  from  it  incorrect  rules  about 
the  impact  of  bodies.  On  the  other  hand,  Wallis  made  successful 
use  of  the  principle  to  discover  the  laws  of  the  transfer  of  motion  in 
the  impact  of  hard  or  elastic  bodies.  And  Lagrange  continues,  "  Just 
as  the  product  of  the  mass  and  the  velocity  represents  the  finite  force 
of  a  body  in  motion,  so  the  product  of  the  mass  and  the  accelerating 
force — which  we  have  seen  to  be  represented  by  the  element  of  velocity 
divided  by  the  element  of  time — will  represent  the  elementary  or  nascent 
force.  Ajid  this  quantity,  if  it  is  considered  as  the  measure  of  the  effect 
that  the  body  can  exert  because  of  the  velocity  which  it  has  assumed, 
or  which  it  tends  to  assume,  constitutes  what  is  called  pressure  ;  but 
if  it  is  regarded  as  a  measure  of  the  force  or  power  necessary  to  impart 
this  same  velocity,  it  is  then  what  is  called  motive  force.  " 

In  modern  language,  the  finite  force  of  a  body  in  motion  is  represent 
ed  by  the  product  mv,  and  the  "  elementary  or  nascent  force " 
u  * 

bym5T 

Lagrange  does  not  openly  take  sides  between  Euler's  thesis — 
based  on  the  law  Fdt  =  mdv  (where  F  is  the  static  force) — and  d'Alem- 
bert's  thesis.  This  matter  of  principle  interested  him  less  than  the 
formal  organisation  of  dynamics,  which  was  the  primary  object  of 
his  own  treatise.  Because  of  the  work  of  his  predecessors,  the  mechanics 
of  a  particle  had  no  mystery  for  him.  Primarily,  he  sought  to  provide 
statics,  and  then  the  dynamics  of  systems,  with  the  general  method 
that  they  still  lacked. 

In  turn,  Lagrange  analyses  the  four  principles  of  dynamics —  the 
conservation  of  living  forces  ;  the  conservation  of  the  motion  of  the 
centre  of  gravity ;  the  conservation  of  moments  or  the  principle  of  areas  ; 
and  the  principle  of  the  least  quantity  of  action. 

Lagrange  says,  legitimately,  that  the  first  of  these  principles  goes 
back  to  Huyghens  "  in  a  form  a  little  different  from  that  in  which  it 

1  Mfaanique  analytique,  Vol.  I,  p.  213. 


LAGRANGE  341 

is  presented  now.  "  Jean  Bernoulli,  following  Leibniz,  fashioned  it 
into  the  principle  of  the  conservation  of  living  forces.  Daniel  Bernoulli, 
after  applying  it  to  fluids,  extended  it  (in  the  Memoires  of  Berlin  for 
1748)  to  a  system  of  bodies  attracting  each  other,  or  tending  towards 
fixed  centres,  according  to  any  law  which  is  a  function  of  distance. 

The  second  principle  is  due  to  Newton  and  was  revived  by  d'Alem- 
bert. 

The  third  principle,  discovered  by  Euler,1  Daniel  Bernoulli,2 
and  d'Arcy,3  is  only  the  generalisation  of  a  theorem  of  Newton 
concerning  several  particles  attracted  by  the  same  centre. 

D'Arcy  went  further  and  sought  to  make  the  principle  of  areas 
into  a  principle  of  the  conservation  of  action.  Lagrange  protests, 
66  As  if  this  vague  and  arbitrary  nomenclature  were  the  essence  of  the 
laws  of  nature  and  could,  by  some  secret  property,  elevate  the  simple 
results  of  the  known  laws  of  mechanics  into  final  causes.  "  4 

The  criticism  which  Lagrange  directs  against  Maupertuis'  principle 
merits  quotation. 

"  Finally  I  come  to  the  fourth  principle,  which  I  call  that  of  least 
action  by  analogy  with  that  which  Maupertuis  gave  under  the  same 
name,  and  which  the  writings  of  many  illustrious  authors  have  since 
made  so  well-known.  This  principle,  looked  at  analytically,  consists 
in  that,  in  the  motion  of  bodies  which  act  upon  each  other,  the  sum 
of  the  products  of  the  masses  with  the  velocities  and  with  the  distances 
travelled  is  a  minimum.  The  author  deduced  from  it  the  laws  of 
the  reflection  and  refraction  of  light,  as  well  as  those  of  the  impact 
of  bodies. 

"  But  these  applications  are  too  particular  to  be  used  for  esta 
blishing  the  truth  of  a  general  principle.  Besides,  they  have  a  somewhat 
vague  and  arbitrary  character,  which  can  only  render  the  conclusions 
that  might  have  been  deduced  from  the  true  correctness  of  the  principle 
unsure.  Further,  it  seems  to  me  that  it  would  be  wrong  to  place 
this  principle,  presented  in  this  way,  among  those  which  we  have 
just  given.  But  there  is  another  way  in  which  it  may  be  regarded, 
more  general,  more  rigorous,  and  which  itself  merits  the  attention 
of  the  geometers.  Euler  gave  the  first  hint  of  this  at  the  end  of  his 
Traite  des  isoperimetres,  printed  at  Lausanne  in  1744.  He  demon 
strated,  in  the  trajectories  described  under  the  action  of  central  forces, 
that  the  integral  of  the  velocity  multiplied  by  the  element  of  the  curve 

1  Opuscules,  Vol.  I,  1746. 

2  Memoires  de  Berlin,  1746. 

3  Memoires  de  FAcademie  des  Sciences,  1747. 

4  Mecanique  analytigue,  Vol.  I,  p.  228. 


342  THE   PRINCIPLES  OF  CLASSICAL   MECHANICS 

is  always  a  maximum  or  a  minimum.  By  means  of  the  conservation 
of  living  forces  I  have  extended  this  property,  which  Euler  discovered 
in  the  motion  of  isolated  bodies  and  which  seemed  confined  to  these 
bodies,  to  the  motion  of  any  system  of  bodies  which  interact  in  any 
way.  From  this  has  come  a  new  general  principle,  that  the  sum  of 
the  products  of  the  masses  with  the  integrals  of  the  velocities,  each 
of  which  is  multiplied  by  the  element  of  distance  travelled,  is  invariably 
a  maximum  or  a  minimum. 

"  This  is  the  principle  which  I  now  give,  however  improperly, 
the  name  of  least  action.  I  regard  it  not  as  a  metaphysical  principle, 
but  as  a  simple  and  general  result  of  the  laws  of  mechanics.  "  x 


4.  LAGRANGE  's  EQUATIONS, 

Lagrange  was  able  to  put  the  equations  of  dynamics  into  a  very 
general  and  valuable  form  which  has  now  become  classical. 

For  each  element,  of  mass  m,  of  a  system,  Lagrange  defines  "  the 
forces  parallel  to  the  axes  of  coordinates  which  are  used,  directly, 
to  move  it,  "  to  be 

d*x  d*y  d*z 

~ 


He  regards  each  element  of  the  system  as  acted  upon  by  similar 
forces,  and  concludes  that  the  sum  of  the  moments  2  of  these  forces 
must  always  be  equal  to  the  sum  of  the  given  accelerating  forces  which 
act  on  each  element.  Thus  he  writes 

Rdr  +  ...)  =  0 

the  given  forces  P,  (),  JR,  .  .  .  being  supposed  to  act  on  each  element 
along  the  lines  p,  £,  r,  .  .  . 

Lagrange  transforms  the  first  sum  by  using  the  identity 

d*xdx  +  d*ydy  +  dzzdz  =  d  (dxdx  +  dydy  +  dzdz)  —  -6  (dx*  +  dy*  +  dz*}. 

By  a  change  of  variables  in  which  each  differential  dx,  dy,  dz,  .  .  . 
is  expressed  as  a  linear  function  of  the  differentials  d£,  dip?  d<p,  .  .  .  , 

1  Mecanique  analytigue,  Vol.  I,  pp.  229,  230. 

2  In  the  sense  already  encountered  in  LAGRANGE'S  statics. 


IAGRANGE  343 

Lagrange  establishes  that  if  0  is  the  transform  of  the  quantity 

i  (da*  +  dy*  +  dz*) 

£ 

then  the  following  equation  is  identically  true. 


Lagrange  confines  himself  to  forces  P,  Q,  R,  .  .  .  for  which  the 
quantity 

Pdp  +  Qdq  +  Rdr  + 

is  integrable,  which,  he  declares,  "  is  probably  true  in  nature.  "     This 
enables  him  to  suppose  that 

Sm  (Pdq  +  Qdq  +  Rdr  +...)  =  d  SroZT(f,  y9  (p.  .  .). 
The  general  equations  of  dynamics  are  then  written  in  the  form 

sdf  +  y%  +  .  .  .  -  o 

by  putting 

~,3T       dT      dV 


with 

*       d*       dzz 


Having  arrived  at  these  results,  Lagrange  examines  the  particularly 
interesting  circumstance  in  which  the  variables  f,  yj,  .  .  .  are  exactly 
sufficient  to  characterise  the  motion  of  the  system  after  all  the  equations 
of  constraint  have  been  eliminated. 

66  If,  in  the  choice  of  the  new  variables  £,  ^,  .  .  .  ,  regard  has  been 
paid  to  the  equations  of  condition  provided  by  the  nature  of  the  proposed 
system,  so  that  the  variations  are  now  completely  independent  of 
each  other  and  that,  consequently,  their  variations  <5£,  <5y>,  .  .  .  ,  remain 
absolutely  indeterminate,  then  the  particular  equations 


will  serve  to  determine  the  motion  of  the  system,  since  these  equations 
are  equal  in  number  to  the  variables  £,  ^,  .  .  .  on  which  the  position 
of  the  system  at  each  instant  depends.  "  1 

1  Mecanique  analytique,  Vol.  I,  p.  291. 


344  THE   PRINCIPLES  OF  CLASSICAL  MECHANICS 

Lagrange  connects  this  analysis  with  the  method  of  multipliers 
which  he  introduced  in  statics.  If  the  variables  f,  ip,  .  .  .  are  greater 
in  number  than  the  degrees  of  freedom  of  the  system,  they  will  be 
related  by  the  equations 

L  =  0        M=  0        N=  0  ... 
Then  Lagrange's  general  formula  becomes 

Sdg  +  Wdy  +  ...  +  16L  +  pdM  +  vdN  +  . . .  =  0 
whence  the  equations  of  motion 

^          dL  dM         dN 

aL       BM 

t    -J-  A.  ~r—   -p  U  -r —  • 

oip  oyj 

which  must  be  associated  with  the  equations  of  constraint. 

The  method  of  multipliers,  which  Lagrange  himself  only  applied 
here  to  the  systems  of  constraints  which  are  now  called  holonomic, 
is  easily  extended  to  non-holonomic  constraints — that  is,  to  constraints 
which  cannot  be  expressed  finitely  as  functions  of  f,  ip,  .  .  . 

5.    THE  CONSERVATION  OF  LIVING  FORCES  AS  A  COROLLARY  OF  LAGRANGE *S 
EQUATIONS. 

Better  than  d'Alembert  had  been  able  to  do,  Lagrange  established 
that  the  conservation  of  living  forces  is  a  consequence  of  the  equations  of 
dynamics,  as  long  as  the  constraints  are  without  friction  and  independent 
of  time. 

For  this  purpose,  Lagrange  considers  the  true  motion  of  the  system 
between  the  time  t  and  the  time  t  +  dt ;  that  is,  he  substitutes  dx, 
Ay,  dz,  ..  .  and  dp,  dq,  dr, . .  .  for  dx,  6y,  6z,  .  .  .  and  dp,  dq,  dr, .  .  .  in 
the  general  formula.  This  enables  him  to  write 

Q      fdxd2x  H 

U77I 


If  the   quantity 

Pdp  +  Qdq  +  Rdr  +  ... 
is  integrable,  then 


LAGRANGE  345 

"  This  equation  includes  the  principle  known  by  the  name  of  the 
conservation  of  living  forces.  Indeed,  since  dx*  +  dy2  +  &&  is  the 
square  of  the  distance  which  the  body  travels  in  the  time  eft,  then 

x      -. — — — ! — —  will  be  the  square  of  the  velocity  and  m ~ 

dt2  dtr 

will  be  its  living  force.     Therefore 

c  fdx*  +  dy2  +  • 


3? 


771 


will  be  the  living  force  of  the  whole  system,  and  it  is  seen,  by  means 
of  the  equation  concerned,  that  this  living  force  is  equal  to  the  cpian- 
tity  2H  —  2  Slim,  which  only  depends  on  the  accelerating  forces 
which  act  on  the  bodies  and  not  on  their  mutual  constraints.  So 
that  the  living  force  is  always  the  same  as  that  which  the  bodies 
would  have  acquired  if  they  had  moved  freely,  each  along  the  line 
that  it  described,  under  the  influence  of  the  same  powers.  "  z 

Thus  Lagrange  discovers  the  same  principle  as  that   formulated 
by  Huyghens  to  be  a  simple  corollary  of  his  general  equations. 


6.    THE   PRINCIPLE   OF   LEAST   ACTION  AS   A  COROLLARY  OF  LAGRANGE's 
EQUATIONS. 

Lagrange  starts  from  the  equation  of  living  forces 

d#2  +  dy2  +  A 


and  differentiates  it  to  obtain 

STTI  (udu  +  dll)  =  0 
or 

Sm  (Pdp  +  Qdq  +  Rdr  +...)  =  —  Smudu. 

Substitution  in  the  general  formula  leads  to 


or 

Sm    7^; u2  -j-  —  udu    =  0 


Mecanique  analytique,  Vol.  I,  p.  268. 


346  THE   PRINCIPLES   OF  CLASSICAL   MECHANICS 

or  again 


[ 


*(*** 


dt 
and  finally 

oC      f     7        c     fdx        .   dy  dz     \ 

oom     uds  =  om  I  -=-  ox  +  -=r-  or  4-  -=-  oz    . 
J  \<ft         {   dt    J       dt      / 

If  it  is  supposed  that  the  variations  <5#,  <5y,  dz  are  zero  at  the  ends  of 
the  ranges  of  integration,  then 

<5S  J  muds  =  0, 

Lagrange  concludes,  "  In  the  general  motion  of  any  system  of  bodies, 
actuated  by  mutual  forces  of  attraction,  or  by  attractions  towards 
fixed  centres  which  are  proportional  to  any  function  of  the  distance, 
the  curves  described  by  the  different  bodies,  and  their  velocities,  are 
necessarily  such  that  the  sum  of  the  products  of  each  mass  by  the 
integral  of  the  product  of  the  velocity  and  the  element  of  the  curve 
is  necessarily  a  maximum  or  a  minimum  ;  provided  that  the  first  and 
last  points  of  each  curve  are  regarded  as  fixed,  so  that  the  velocities 
of  the  corresponding  coordinates  at  those  points  are  zero.  "  l 

Maupertuis'  principle  is  thus  found  to  be  valid,  in  a  more  general 
form  than  that  which  Euler  gave  it.  Moreover,  this  principle  expresses 
the  extremal  character  of  the  living  force  between  two  known  confi 
gurations  of  the  system.  This  Lagrange  establishes  in  the  following 
way. 

"  Since  ds  —  udt,  the  formula 

Sm     uds 

which  is  either  a  maximum  or  a  minimum,  can  be  put  in  the  form 
Sm     u2dt       or          eftSmu2. 

Here  Smw2  represents  the  living  force  of  the  whole  system  at  any 
time.  Thus  the  principle  reduces  to —  the  sum  of  the  instantaneous 
living  forces  of  all  the  bodies,  from  the  moment  that  they  start  from 
given  points  to  that  when  they  arrive  at  other  given  points,  is  a  maxi 
mum  or  a  minimum.  It  could  be  called,  with  more  justice,  the  prin 
ciple  of  the  greatest  or  least  living  force,  and  this  way  of  regarding  it 
would  have  the  advantage  of  being  general,  since  the  living  force  of 
a  system  is  always  greatest  or  least  in  the  equilibrium  condition.  "  2 

1  Mtcanique  analytique,  Vol.  I,  p.  276. 

2  Ibid.,  p.  281. 


LAGRANGE 


347 


7.    ON   SOME   PROBLEMS   TREATED   IN   THE    "  MECANIQTJE   ANALYTIQUE." 

Mecanique  analytique  includes  a  study  of  a  great  number  of  pro 
blems  which  we  are  not  able  to  treat  in  this  book. 

We  note,  however,  that  Lagrange  initiated  a  general  method  of 
approximation  in  dynamical  problems  which  was  based  on  the  variation 
of  arbitrary  constants  ;  that  he  developed  the  theory  of  small  motions  ; 
that  he  studied  the  stabilty  of  equilibrium,  and  stated  that  equilibrium 
is  necessarily  stable  when  the  potential  of  the  given  forces  is  a  minimum. 
This  demonstration  was  to  be  perfected  by  Lejeune-Dirichlet. 

Lagrange  also  studied  in  detail  the  motion  of  a  heavy  solid  of 
revolution  which  was  suspended  from  a  point  on  its  axis,  and  expressed 
the  solution  in  terms  of  elliptic  integrals. 

8.  LAGRANGE'S  HYDRODYNAMICS, 

After  describing  the  historical  development  of  hydrodynamics, 
Lagrange  made  a  very  important  contribution  to  the  subject. 

As  a  supplement  to  Euler's  variables,  Lagrange  introduced  the 
variable  with  which  his  name  is  still  associated  into  the  kinematics 
of  continuous  media.  The  actual  coordinates  of  an  element  of  the 
medium  are  considered  as  functions  of  the  time  and  of  the  initial 
coordinates  a,  6,  c,  of  the  same  element. 

Lagrange  established  a  fundamental  theorem  on  the  permanence 
of  the  irrotational  property  in  fluid  motion. 

If  the  fluid  is  first  supposed  to  be  incompressible  and  homogeneous, 
and  its  density  is  taken  equal  to  unity,  Lagrange1  also  assumed  that 
the  accelerating  forces  X,  Y,  Z  which  act  on  the  elements  of  the  fluid  are 
such  that  Xdx  +  Ydy  +  Zd*  is  an  exact  differential  dV.  Lagrange  writes 

fdu  du  du  ,  du 


dv    ,       dv 

s-x+v^ 

Sw  ,      dw 


The  right-hand  side  of  this  equation,  like  the  left  hand  side,  must 
be  an  exact  differential     Now  the  right-hand  side  can  be  written  as 


1  Mfaanique  analytique,  Vol.  II,  p.  268. 


348  THE   PRINCIPLES  OF  CLASSICAL   MECHANICS 

Lagrange  remarks  that  this  quantity  will  be  an  exact  differential 
whenever  udx  +  vdy  +  wdz  is  such,  "  but  as  this  is  only  a  special 
supposition,  it  is  necessary  to  inquire  in  what  cases  it  can  and  must 
be  appropriate.  " 

Lagrange  then  verifies  that  when  u,  v,  w  are  expanded  as  functions 
of  the  time,  in  the  form 


V  =    .  .  .       W  = 


it  is  necessary  that,  whenever  udx  +  v'dy  +  w'dz  is  an  exact  differential, 

that  u"dx  +  v"dy  +  w"<fc,  u'"dx  +  v'"dy  +  w'"dx,  etc should 

also  be  exact  differentials. 

He  concludes,  "  From  this  it  follows  that  if  the  quantity  udx  + 
vdy  +  wdz  is  a  total  exact  differential  when  t  =  0,  it  must  also  be 
a  total  exact  differential  when  t  has  any  other  value.  Therefore, 
in  general,  since  the  origin  of  t  is  arbitrary,  and  since  t  can  either  be 
taken  positive  or  negative,  it  follows  that  if  the  quantity  udx  +  vdy  + 
wdz  is  a  total  exact  differential  at  any  time,  it  must  be  such  at  all  other 
times. 

"  Accordingly,  if  there  is  a  single  instant  at  which  it  is  not  a  total 
exact  differential,  it  can  never  become  such  throughout  the  motion. 
For  if  it  were  a  total  exact  differential  at  any  instant,  it  would  also 
be  such  at  the  first.  " 

This  theorem  of  Lagrange  is  a  fine  example  of  discovery  achieved 
by  a  procedure  which  appears  to  be  purely  mathematical. 

In  sympathy  with  the  spirit  of  the  time,  which  readily  assumed 
that  nature  conformed  to  simple  laws,  Lagrange  declared  "  that  it 
is  possible  to  ask  whether  there  are  motions  for  which  udx  +  vdy  + 
wdz  is  not  a  total  exact  differential.  " 

To  answer  this  question,  he  shows  that  in  the  motion 

u  =  gy         v  =  — •  gx         w  =  0 

the  condition  that  udx  +  vdy  +  wdz  should  be  a  complete  differential 
is  not  satisfied,  although  it  is  possible  to  write 

p  =  V-  —  (*2  +  y2)  +  funct.  t. 

2t 

Now  "  it  is  clear  that  these  values  of  u,  v,  w  represent  the  motion 
of  a  fluid  which  rotates  with  a  constant  angular  velocity  equal  to  g 
about  the  fixed  axis  of  coordinates  z.  And  it  is  known  that  such  a 
motion  can  always  take  place  in  a  fluid.  From  this  it  can  be  concluded 
that  in  the  oscillations  of  the  sea  due  to  the  attraction  of  the  Sun  and 


LAGRANGE  349 


the  Moon,  it  cannot  be  supposed  that  the  cpiantity  udx  +  vdy  -j- 
is  integrable,  since  it  is  not  so  when  the  fluid  is  at  rest  with  respect 
to  the  Earth  and  only  has  the  same  rotational  motion  as  the  Earth.  " 
Lagrange  extended  his  theorem  to  compressible  fluids  by  intro 
ducing  an  "  elasticity  "  that  was  a  function  of  density  alone,  so  that 

—  is  the  differential  of  some  function  E(Q). 

e 

We  also  mention  Lagrange's  study  of  the  motion  of  a  fluid  in  an 
almost  horizontal  shallow  canal.  This  motion  is  governed  by  an 
equation  similar  to  the  equation  of  the  propagation  of  sound.  The 
wave  velocity  turns  out  to  be  proportional  to  the  s<juare  root  of  the 
depth  of  the  fluid  if  the  canal  has  a  uniform  breadth. 


PART  FOUR 

SOME  CHARACTERISTIC  FEATURES 

OF  THE  EVOLUTION 
OF  CLASSICAL  MECHANICS  AFTER  LAGRANGE 


FOREWORD 


It  would  seem  that  the  valuable  function  of  history  is  that  analysing 
the  paths  which  scientific  thought  has  travelled  in  the  creation,  in  a 
limited  field  like  that  of  classical  mechanics,  of  a  rationally  organised 
structure. 

The  material  for  this  study  thus  consist  of  the  vicissitudes  en 
countered  on  these  paths,  the  interaction  between  currents  of  thought 
which  were  in  principle  divergent,  or  even  opposed.  It  is  difficult 
material,  sometimes  deceptive,  but  always  revealing  of  the  profound 
difficulties  of  research  and — for  this  reason — instructive. 

After  Lagrange,  after  the  efforts  of  the  students  of  the  XVIIIth 
Century,  mechanics  had  attained  this  rational  structure.  It  lasted 
until  the  impact  of  the  needs  of  relativistic  and  quantum  physics. 

The  intervening  period  was  a  didactic  one,  which  we  are  reluctant 
to  deal  with  for  fear  of  duplicating  the  books  that  have,  rightly,  become 
classical. 

That  is  why,  in  the  following  pages,  we  shall  confine  ourselves 
to  some  characteristic  features  of  the  evolution  of  classical  mechanics 
after  Lagrange.  We  shall  be  concerned  with  discussions  of  the  prin 
ciples  themselves,  and  with  certain  isolated  facts  to  which  it  seems 
natural  to  attach  a  historical  significance  because  of  their  influence 
on  the  later  development  of  mechanics. 


23 


CHAPTER   ONE 


LAPLACE'S  MECHANICS  (1799) 


1.  LAPLACE  AND  THE  PRINCIPLES   OF  DYNAMICS. 

We  shall  discuss  here  only  that  part  of  Laplace's  work  which  is 
directly  concerned  with  the  principles  of  dynamics. 

Laplace  referred  the  motion  of  bodies  to  "  an  infinite  space,  at 
rest  and  penetrable  to  matter.  "  x 

The  concept  of  force  evoked  the  following  comment. 

"  The  mechanism  of  that  remarkable  agency,  force,  by  which  a 
body  is  moved  from  one  place  to  another,  is  and  always  will  be  unknown. 
It  is  only  possible  to  determine  the  laws  which  govern  its  behaviour. 
A  force  acting  on  a  particle  will  necessarily  set  it  in  motion,  if  there 
is  nothing  to  prevent  this.  The  direction  of  the  force  is  that  of  the 
straight  line  which  the  particle  is  made  to  describe.  "  2 

Laplace  defined  inertia  as  the  tendency  of  matter  to  remain  in 
its  state  of  rest  or  motion.  That  the  direction  of  motion  was  constant 
appeared  obvious  to  him  ;  with  respect  to  its  uniformity,  he  pointed 
out  that  the  law  of  inertia  was  the  simplest  conceivable  law,  and  that 
it  was  justified  by  astronomical  and  terrestrial  observations. 

The  Laplace  endeavoured  to  prove  that  "  force  "  is  proportional 
to  the  velocity.  He  is  concerned  here,  of  course,  with  the  force  of 
a  moving  body. 

Let  v  be  the  velocity  of  the  Earth,  common  to  all  bodies  on  its 
surface,  and  /  be  the  force  which  a  particle  M  experiences  because 

of  this  motion.    The  ration-  is  an  unknown  function  of  /,   say   <p(f). 

The  form  of  this  function  has  to  be  found  by  a  method  which  has 
recourse  to  experimental  observation. 

Suppose  that  M  is  also  acted  upon  by  another  force,  /',  which 

1  Mecanique  celeste,  Part  I,  Book  I,  p.  3  (1799). 

2  Ibid.,  p.  4. 


LAPIACE  355 

combines  with  f  to  form  a  resultant  F  according  to  the  parallelogram 
rule.  Under  these  conditions  the  particle  will  acquire  a  certain  velocity 
U. 

Laplace  now  argued  that  /'  could  be  considered  as  "  infinitely 
small  "  compared  with  /.  "  The  greatest  forces  that  we  are  able  to 
impress  on  bodies  on  the  surface  of  the  earth  are  much  smaller  than 
those  that  it  experiences  because  of  the  motion  of  the  earth.  " 

Accordingly,  the  relation 


yields 


J 

if  /'2  is  neglected  in  comparison  with  /.     It  follows  that 

g>  (F)  =  <p  (/)  +  f-f  <p'(f)    with    ?'(/)  =  & 
J  dt 

The  relative  velocity  of  M  with  respect  to  the  Earth,    U  —  v, 
equal  to 

F,p(F)-f<p(f) 
which  can  easily  be  shown  to  lead  to  the  equation 


is 


From  this  it  follows,  in  the  general  case  in  which  the  directions  of  f 
and  f  are  not  the  same,  that  this  relative  velocity  must  have  a  compo 
nent  perpendicular  to  that  of  the  impressed  force/'  unless  <p'(f)  vanishes. 
(It  is  assumed  that  the  scalar  product  /  •  /'  is  not  zero.) 

At  this  point  Laplace  appeals  to  experiment. 

"  Thus,  imagine  that  a  sphere  at  rest  on  a  smooth  horizontal  plane 
is  hit  by  the  base  of  a  right  cylinder,  moving  along  the  direction  of 
its  axis  which  is  supposed  to  be  horizontal.  The  apparent  relative 
motion  of  the  sphere  should  not  be  parallel  to  this  axis,  for  all  positions 
of  the  axis  with  respect  to  the  horizon.  Here  is  a  simple  means  of 
finding  out  by  experiment  whether  <p'(f)  has  an  appreciable  value 
on  the  Earth.  But  the  most  accurate  experiments  do  not  demonstrate 
any  deviation  of  the  apparent  motion  of  the  sphere  from  the  direction 
of  the  impressed  force.  From  which  it  follows  that  on  the  Earth 
<p'(f)  is  very  nearly  equal  to  zero.  Its  value,  however  inappreciable 
it  might  be,  would  make  itself  apparent  in  the  length  of  the  oscillations 


356  THE   EVOLUTION   OF    CLASSICAL 

of  a  pendulum,  which  would  vary  according  to  the  position  of  the 
plane  of  its  motion  with  respect  to  the  direction  of  the  Earth's  motion. 
The  most  accurate  observations  do  not  reveal  any  such  difference. 
We  must  then  conclude  that  <p'(f)  is  inappreciable,  and  can  be  supposed 
to  be  zero,  on  the  Earth. 

"  If  the  equation  q>'(f)  =  0  was  obtained  whatever  the  force /might 
be,  <p(f)  would  be  constant  and  the  velocity  would  be  proportional 
to  the  force.  It  would  also  be  proportional  to  the  force  if  the 
function  <p(f)  were  only  composed  of  a  single  term,  since  otherwise 
V'(f)  would  never  be  zero.  Therefore,  if  the  velocity  were  not  propor 
tional  to  the  force,  it  would  be  necessary  to  suppose  that  in  nature 
the  function  of  the  velocity  which  represents  the  force  consists  of 
several  terms,  which  is  unlikely.  It  would  also  be  necessary  to  suppose 
that  the  velocity  of  the  Earth  is  exactly  that  which  the  equation  <p'(f)  —  0 
requires,  which  is  against  all  probability.  Moreover,  the  velocity 
of  the  Earth  varies  in  the  different  seasons  of  the  year — it  is  about  a 
thirtieth  greater  in  winter  than  in  summer.  This  variation  is  still 
more  considerable  if,  as  everything  seems  to  show,  the  solar  system 
is  in  motion  through  space — for  whether  this  progressive  motion 
combines  with  that  of  the  Earth  or  whether  it  is  opposed  to  it,  during 
the  course  of  the  year  there  must  result  large  variations  in  the  absolute 
motion  of  the  Earth.  This  would  modify  the  equation  concerned 
and  thus  the  relation  of  the  impressed  force  to  the  absolute  velocity, 
if  this  equation  and  this  velocity  were  not  independent  of  the  motion 
of  the  Earth.  However,  observation  has  not  revealed  any  appreciable 
variation.  "  1 

Laplace  concludes,  "  Here  then  are  two  laws  of  motion  ;  namely, 
the  law  of  inertia  and  that  according  to  which  the  force  is  proportional 
to  the  velocity,  and  these  are  provided  by  observation.  They  are 
the  simplest  and  most  natural  that  could  be  imagined  and  undoubtedly 
they  derive  from  the  nature  of  matter  itself.  But  since  this  nature 
is  unknown  they  are  merely,  for  us,  observed  facts  ;  moreover,  the 
only  ones  which  mechanics  borrows  from  observation.  "  2 

Laplace  next  gave  his  attention  to  "  forces  which  appear  to  act 
in  a  continuous  manner,  like  gravity.  "  Here,  like  Carnot,  Laplace 
considers  that  gravity  acts  in  successive  impulses  at  infinitely  small 
intervals  of  time.  "  We  suppose  that  the  interval  of  time  which 
separates  two  actions  of  some  force  is  equal  to  the  element  of  time  dt. 
It  is  clear  [that  the  instantaneous  action  of  the  force]  must  be  supposed 
to  be  proportional  to  the  intensity  and  to  the  element  of  time  in  which 

1  Mecanique  celeste,  Part  I,  Book  I,  p.  17. 

2  Ibid.,  p.  18. 


LAPLACE  357 

it  is  supposed  to  act.  Thus,  representing  the  intensity  by  P,  it  must 
be  supposed,  at  the  beginning  of  each  instant  A,  that  the  particle 
is  actuated  by  a  force  Pdt  and  that  it  is  moved  uniformly  during  that 
instant.  "  l 

2.    THE  GENERAL  MECHANICS  COMPATIBLE  WITH  AN  ARBITRARY  RELATION 
BETWEEN   THE    "  FORCE  "   AND   THE   VELOCITY. 

In  the  last  §  we  have  seen  Laplace  emphasise  the  communication 
of  motion  without  seeking  to  elucidate  the  original  causes  of  the  motion. 
He  thus  belongs  to  the  tradition  of  d'Alembert  and  Carnot.  In  parti 
cular,  like  Carnot,  he  asserted  the  experimental  character  of  the  laws 
of  mechanics.  His  analysis  —  this  time  entirely  original  —  of  the  notion 
of  the  force  of  a  body  in  motion  led  him,  by  an  innate  propensity  to 
purely  mathematical  generalisation,  to  an  extension  of  dynamics 
which  encompassed  the  ideas  that  the  physicists  were  to  use,  a  century 
later,  in  special  relativity. 

This  extension  was  the  subject  of  Chapter  VI  of  the  first  part 
of  Book  I  of  the  Mecanique  celeste.  The  chapter  is  called  "  On  the 
laws  of  motion  of  a  system  of  bodies  associated  with  all  possible  mathemat 
ical  relationships  between  the  force  and  the  velocity.  " 

Laplace  remarks  (as  we  have  mentioned  in  the  preceding  paragraph) 
that  there  is  an  infinite  number  of  self-  consistent  ways  of  expressing 
the  "  force  "  in  terms  of  the  velocity.  This  infinity  corresponds  to 
all  possible  forms  of  the  relation  between  the  force  and  the  velocity. 

F=<p(v) 
The  general  equation  of  the  dynamics  of  systems  is 


and  is  valid  when  the  "  force  "  is  proportional  to  the  velocity.  In 
order  to  obtain  the  generalisation  which  is  sought,  it  is  sufficient  to 
assume  that  the  body  of  mass  m  is  actuated  by  a  "  force  "  whose 
components  parallel  to  the  axes  are 

doc  dy  dz 

*w2s       *(v]Ts       *(v]ds- 

In  the  instant  following  this  force  becomes 

.  .  dx   ,     *  /    ,  .    dx\ 
9(t))_  +  d^(v)._j    etc. 

1  Mtcanique  celeste.  Part  I,  Book  I,  p.  19. 


358  THE    EVOLUTION   OF    CLASSICAL 

,  .  dx 
f^dTS  + 

Then  the  general  equation  of  the  dynamics  of  systems  takes  the 
form 


*)-"*)}-* 


Here  —  ,  -j-  and  -—  appear  as  products  with  the  function  —  —  ,  which 
at     at  at  v 

reduces  to  unity  "  for  that  natural  law  according  to  which  the  force 
is  proportional  to  the  velocity.  " 

Laplace  remarked  that  this  difference  makes  the  solution  of  the 
problems  of  mechanics  very  difficult. 

But  the  principle  of  the  conservation  of  living  forces,  the  principle 
of  areas  and  the  principles  of  the  centre  of  gravity  and  of  least  action, 
can  be  extended  to  this  case. 

The  extension  of  the  principle  of  living  forces  is  obtained  by  sub 
stituting  dx9  dy  and  dz  for  6x,  dy  and  dz  in  the  general  equation.  Thus, 
if  U  is  the  function  of  the  forces, 


£  J  mvdv-(p'(v)  =U  +  h      with      <p'(v)  = 


"  The  principle  of  the  conservation  of  living  forces  therefore  obtains 
for  all  the  possible  mathematical  relationships  between  the  force  and 
the  velocity,  provided  that  the  living  force  of  a  body  is  understood 
as  the  product  of  its  mass  and  twice  the  integral  of  the  product  of 
its  velocity  and  the  differential  of  the  function  of  velocity  which 
represents  the  force.  " 

In  the  same  way,  Laplace  extended  the  theorem  of  quantities 
of  motion  to  an  isolated  system.  He  generalised  the  principle  of  areas 
to  the  form 

fxdy  —  ydx\     cp  (v) 
m        J       J  —     .  TJU.  —  constant. 
\        dt        )        v 

Finally,  he  wrote  the  generalised  principle  of  least  action  as 


Thus,  as  early  as  1799,  Laplace  was  able  to  formulate  the  general 
mechanics   of  which  the  dynamics  of  special  relativity,  in  a  given 


LAPLACE  359 

reference  system,  is  only  a  particular  case.  However,  there  is  a  slight 
difference  of  meaning  between  Laplace's  purely  mathematical  concep 
tion  and  that  of  the  modern  physicists.  To  Laplace,  the  mass  m  of 
a  particle  remained  constant  and  it  was  the  momentum  which  was 
no  longer  proportional  to  the  velocity.  In  the  physical  theory  of 
relativity,  on  the  other  hand,  the  mass  M  becomes  a  function  of  the 
velocity  while  the  momentum  remains  in  the  form  Mv. 

In  order  to  pass  from  one  of  these  systems  to  the  other,  it  is  suffi 
cient  to  put 

cp(v) 


M  =  m  • 


v 


3.  LAPLACE  AND  THE   SIGNIFICANCE   OF  THE   LAW  OF   UNIVERSAL 
GRAVITATION. 

In  his  Exposition  du  Systeme  du  Monde  Laplace  recalls  that  when 
Newton  formulated  the  principle  of  universal  gravitation,  Descartes 
had  precisely  managed  to  "  substitute  the  intelligible  ideas  of  motion, 
impulse  and  centrifugal  force  for  the  occult  quantities  of  the  Aris 
totelians.  " 1  His  system  of  vortices  met  with  the  approval  of  the 
philosophers,  "  who  rejected  the  obscure  and  meaningless  doctrines 
of  the  Schoolmen,  and  who  believed  that  they  saw  those  occult  features, 
that  French  philosophy  had  so  legitimately  banned,  reborn  in  the 
attractions.  "  2 

Laplace  opined  that  Newton  would  have  deserved  this  reproach  if 
he  had  been  content  to  attribute  the  elliptic  motion  of  the  planets 
and  comets,  the  inequalities  of  the  motion  of  the  Moon,  of  the  terres 
trial  degrees  of  latitude  and  gravity,  to  the  universal  attraction  without 
showing  the  connection  between  this  principle  and  these  phenomena. 
But  the  geometers,  rectifying  and  generalising  Newton's  demon 
strations,  had  been  able  to  verify  the  perfect  agreement  between  the 
observations  and  the  results  of  the  analysis. 

Laplace  regarded  "  this  analytical  connection  of  particular  facts 
with  a  general  fact  "  as  a  properly  constituted  theory.  And  he  flattered 
himself  with  having  obtained  one  in  the  deduction  on  the  effects  of 
capillarity  from  a  short  range  interaction  between  molecules  ;  a  true 
theory,  one  which  expresses  the  rigorous  agreement  of  the  calculation 
and  the  phenomena. 

Here  we  see  portrayed  the  dogma  of  universal  attraction.  However 
— and  this  is  essential — it  is  deprived  of  the  a  priori  character  of  the 

1  P.  377. 

2  P.  S78. 


360  THE   EVOLUTION   OF   CLASSICAL 

assertion  of  a  quality  in  the  sense  that  the  Schoolmen  used.  Although 
it  must  certainly  have  passed  through  his  mind,  Laplace  did  not  use 
the  term  dogma  in  this  connection.  For  his  attitude,  supported  at 
many  points  by  experiment,  was  not  necessarily  of  a  dogmatic  character. 

Laplace  assumed,  moreover,  that  the  following  question  could  be 
asked — "  Is  the  principle  of  universal  gravity  a  primordial  law  of 
nature,  or  is  it  merely  the  general  effect  of  an  unknown  cause  ?  "  l 

Laplace  also  asked  whether  the  propagation  of  the  attraction 
was  instantaneous.  An  attempt  to  explain  the  secular  acceleration 
of  the  Moon's  motion  had  led  him  to  assume  that,  if  the  velocity  of 
propagation  was  finite,  it  must  be  seven  million  times  greater  than 
that  of  light. .  .  .  Thus  he  declared  for  an  instantaneous  propagation. 

Further,  he  wrote,  "  Doubtless  the  simplicity  of  the  laws  of  nature 
must  not  necessarily  be  judged  by  the  ease  with  which  we  appreciate 
them.  But  since  those  that  seem  most  simple  to  us  agree  perfectly 
with  all  the  phenomena,  we  are  well  justified  in  regarding  them  as 
being  rigorous.  " 

We  see  that  Laplace's  attitude  was  a  moderate  one,  and  that, 
to  him,  the  certainty  of  a  natural  law  depended  on  a  kind  of  passage 
to  the  limit  in  the  mathematical  sense  of  the  term. 

1  P.  384. 


CHAPTER   TWO 


FOURIER  AND  THE  PRINCIPLE  OF  VIRTUAL  WORKS  (1798) 


We  owe  to  Fourier  a  demonstration  of  the  principle  of  virtual 
works  which  is  based  on  the  equilibrium  of  the  lever,  and  which  has 
been  used  as  the  basis  of  the  presentations  of  the  principle  that  are 
now  classical.1 

Fourier  borrowed  the  notion  of  virtual  velocity  from  Jean  Bernoulli.2 
On  the  other  hand,  he  called  the  moment  of  a  force  the  product  (change 
of  sign)  of  this  force  with  the  virtual  velocity  of  the  point  to  which 
it  is  applied. 

If  a  point  is  in  equilibrium  under  the  action  of  n  forces,  Fourier 
first  verifies  that  the  total  moment  of  these  forces  is  zero  for  an  arbi 
trary  displacement  of  the  point. 

He  then  seeks  the  total  moment  of  two  equal  and  opposed  forces 
"  applied  at  the  ends  of  a  straight  inflexible  line  "  and  acting  in  the 
direction  of  this  line. 

"  If,  at  first,  the  two  points  at  which  these  forces  act  are  regarded 
as  entirely  free,  and  if  each  of  the  points  is  taken  as  the  fixed  centre 
of  the  force  which  acts  on  the  other,  it  will  be  easy  to  see  that,  since 
the  distance  apart  of  the  points  is  a  function  of  their  coordinates, 
the  virtual  velocity  of  the  first  will  be  equal  to  differential  of  the  distance 
when  the  variation  is  made  with  respect  to  the  coordinates  of  this 
point  alone.  It  will  be  the  same  for  the  second  point.  So  that  the 
total  moment,  which  is  proportional  to  the  sum  of  the  virtual  velocities, 
will  also  be  proportional  to  the  sum  of  the  partial  differentials  which 
represent  these  velocities — that  is,  proportional  to  the  complete  dif 
ferential  of  the  distance  between  the  two  points. 

Thus  the  total  moment  of  the  two  forces  is  zero  if  the  distance 
between  the  two  points  is  constant. 

If  the  two  forces  are  repulsive,  the  total  moment  is  negative  when 

1  Memoire  sur  la  Statique,  contenant  la  demonstration  du  principe  des  vitesses  vir- 
tuelles  et  la  theorie  des  moments,  Journal  de  VBcole  poly-technique,  5th  cakier,  1798. 

2  See  above,  p.  232. 


362  THE   EVOLUTION   OF   CLASSICAL 

the  distance  between  the  two  points  increases  and  positive  when  this 
distance  decreases.  Converse  results  are  obtained  when  the  forces 
are  attractive. 

Fourier  then  considers  "  two  inflexible  (and  perfectly  smooth) 
surfaces  which  resist  each  other  "  and  studies  the  total  moment  of 
their  mutual  reactions  in  any  disturbance  of  the  system.  He  considers 
two  points  on  the  common  normal  to  the  surfaces  at  the  point  of  con 
tact  and  such  that  one  lies  inside  each  surface.  These  two  points 
cannot  be  closer  together  than  they  are  in  the  equilibrium  position. 
So  that  either  their  distance  apart  increases  or  it  does  not  change  when 
the  system  is  disturbed. 

"  The  first  distance  is  the  smallest  of  all  those  which  occur  when 
the  position  of  the  two  surfaces  is  varied  in  such  a  way  that  they  remain 
in  contact  with  each  other.  Since  the  law  of  continuity  is  obeyed, 
it  is  necessary  that  the  differential  should  be  zero.  "  Since  the  total 
moment  of  the  reactions  is  proportional  to  the  variation  of  the  distance 
of  the  two  points  at  which  they  act,  it  therefore  remains  zero,  as  long 
as  the  surfaces  remain  in  contact,  whatever  the  displacement  may  be. 

In  order  to  generalise  these  results  Fourier  observes  that  the  mo 
ments  combine  and  decompose  like  forces  (if  a  solid  body  is  concerned). 

If  a  solid  body  is  considered  to  be  in  equilibrium  under  the  action 
of  n  forces  it  is  established  that  the  total  moment  of  the  n  forces  is 
necessarily  zero.  The  converse  is  also  true. 

Fourier  then  imagines  a  system  of  bodies  to  be  connected  by  in- 
extensible  threads  and  acted  upon  by  any  forces  which  are  such  that 
there  would  be  equilibrium  independently  of  any  external  resistance. 
The  forces  which  act  on  each  body  cancel  each  other  out.  Apart 
from  the  forces  directly  applied  to  the  body,  these  forces  comprise 
the  tension  of  the  threads  between  points  of  this  body  and  points 
of  neighbouring  bodies. 

"  That  is  why,  in  considering  simultaneously  all  the  forces  which 
act  on  all  the  bodies,  their  total  moment  can  be  said  to  be  zero  for 
all  conceivable  disturbances — even  for  those  which  the  presence  of 
the  threads  does  not  allow.  It  is  now  necessary  to  select,  from  among 
these  disturbances,  those  which  satisfy  the  equations  of  condition  ; 
and,  for  these  particular  disturbances,  to  discover  the  value  of  the 
total  moment  of  those  forces  which  are  due  to  the  tensions  alone.  " 

This  value  is  zero.  For  each  of  the  threads  is  acted  upon  by  two 
equal  and  opposed  forces,  and  the  distance  between  the  extremities 
is  constant.  From  this  it  follows  that  the  total  moment  of  the  applied 
forces  alone  is  also  zero. 

If  the  distance  between  the  ends  of  the  threads  does  not  remain 


FOURIER  363 

constant,  it  can  only  become  smaller.  Since  the  forces  of  tension 
tend  to  decrease  this  distance,  the  total  moment  of  the  forces  of  tension 
is  negative.  Therefore,  the  sum  of  the  moments  of  the  applied  forces 
alone  can  only  be  positive  for  disturbances  of  this  kind. 

Fourier  next  considers  an  "  undefined  assembly  of  hard  bodies  " 
whose  shapes  and  dimensions  are  arbitrary  and  which  are  supported 
upon  each  other.  Each  body  is  in  equilibrium  under  the  action  of 
the  forces  which  are  applied  to  it  and  the  resistances  of  neighbouring 
bodies.  If  two  neighbouring  bodies  always  remain  in  contact  during 
the  disturbance  of  the  system,  albeit  at  different  points,  the  moment 
of  their  reactions  is  zero.  It  is  negative  if  the  bodies  happen  to  separate. 

**  In  considering  the  combination  of  all  the  forces  which  act  on  all 
the  bodies,  it  is  certain  that  some  of  the  moments  must  be  zero  for 
all  the  disturbances  which  can  be  imagined — even  for  those  which 
may  be  prevented  by  the  mutual  impenetrability  of  the  solids.  Now, 
for  displacements  compatible  with  the  latter  condition,  the  moment 
of  all  the  forces  of  pressure  is  either  zero  or  negative.  Therefore, 
for  all  the  possible  disturbances,  the  sum  of  the  moments  of  the  applied 
forces  alone  is  either  zero  or  positive  l — it  is  zero  when  the  equations 
which  express  the  condition  that  contact  must  take  place  are  satisfied, 
and  positive  whenever  two  bodies  which  touch  each  other,  or  act 
upon  each  other,  become  entirely  separated.  There  is  no  possible 
disturbance  for  which  the  sum  of  the  moments  can  be  negative." 

Fourier  treats  incompressible  fluids  by  considering  that  their 
different  points  are  subject  to  an  interaction  which  opposes  every 
variation  of  the  distances  between  the  points. 

He  then  proceeds  to  the  logical  reduction  of  the  theorem  of  virtual 
work  to  the  principle  of  the  lever.  For  this  purpose  he  replaces  the 
system  by  "  a  simpler  body  which  can  nevertheless  be  disturbed  in 
the  same  way.  " 

Let  p,  j,  r,  s, .  .  .  be  the  points  of  the  given  system  to  which  the 
forces  P,  (),  jR,  S,  .  .  .  are  applied.  The  displacement  which  gives 
the  points  p,  g,  r,  s,  .  *  .  the  initial  virtual  velocities  dp,  dq,  dr,  ds,  .  .  . 
in  the  directions  of  the  lines  p',  q\  r',  s',  .  .  .  is  considered.  The  body 
substituted  for  the  system  will  also  pass  through  the  points  p,  g,  r,  s,  .  .  . 
Similarly,  its  elements  will  have  virtual  velocities  dp,  dg,  dr,  ds,  .  .  . 
along  p',  9',  r',  s',  .  .  . 

Fourier  draws  a  plane  perpendicular  to  the  line  p',  and  passing 
through  the  point  p  ;  also  a  plane  perpendicular  to  qf  through  the  point 

1  Since  FOURIER'S  moment  is,  apart  from  a  change  of  sign,  the  modern  virtual  work, 
this  conclusion  is  the  one  which  is  usually  expressed  in  the  form,  "  The  virtual  work  of 
given  forces  is  zero  or  negative  for  every  displacement  compatible  with  the  constraints.  " 


364  THE    EVOLUTION   OF    CLASSICAL 

q.  These  two  planes  intersect  in  the  straight  line  d.  A  perpendicular, 
ft,  is  dropped  from  p  to  d.  At  the  intersection  of  h  and  d,  and  in  the 
plane  perpendicular  to  q'  passing  through  5,  the  line  hr  is  drawn  per 
pendicular  to  d.  From  q  the  line  h"  is  drawn  perpendicular  to  ft'. 
The  straight  lines  ft  and  h'  are  considered  as  the  arms  on  an  angular 
lever  with  axis  d.  The  straight  line  h"  is  considered  as  a  straight 
lever  with  axis  d  (in  the  plane  perpendicular  to  q'  drawn  through  q). 
Ifp  is  displaced  along  p'  (by  dp)  the  end  of  the  arm  h'  is  correspond 
ingly  displaced,  and  the  axis  d  can  evidently  be  chosen  in  such  a  way 
that  the  displacement  of  q  (required  by  the  straight  lever)  is  exactly 
equal  to  dq.  An  assembly  on  analogous  levers  can  be  imagined  bet 
ween  the  point  q  and  the  point  r,  between  the  point  r  and  the  point  s,  .  .  . 
so  that  the  system  of  levers  thus  constructed  is  susceptible  of  the 
same  displacements  as  those  attributed  to  the  original  system,  and 
of  this  displacement  alone. 


Fig.  107 

Suppose  that  the  forces  P,  (),  .R,  S,  .  .  .  have  a  total  moment 
which  is  zero  for  the  displacement  dp,  dg,  dr,  ds,  .  .  .  Because  of  the 
principle  of  the  lever  and  the  principle  of  the  composition  of  forces, 
the  forces  P,  (),  jR,  S,  .  .  .  will  necessarily  produce  equilibrium  in  the 
system  of  levers  constructed  in  the  way  that  has  been  described. 

We  shall  show,  by  reductio  ad  absurdum,  that  these  same  forces 
will  leave  the  original  system  in  equilibrium.  Indeed,  if  the  points 
p,  g,  r,  s,  .  .  .  assume  the  velocities  dp,  dg,  dr,  ds,  and  if  it  is  assumed 


FOURIER  365 

that  the  point  p  of  the  system  of  levers  is  connected  with  the  point  p 
of  the  given  system,  the  assembly  of  levers  will  be  carried  along  in 
the  displacement  of  the  given  system,  and  the  points  q  and  g,  r  and  r,  .  .  . 
of  the  two  systems  will  not  separate. 

Therefore  it  can  be  supposed  that  there  are  connected  not  only 
the  points  p  and  p,  but  also  the  pairs  of  points  q  and  <jf,  r  and  r,  s  and  5,  ... 
in  the  two  systems.  Accordingly  the  forces  P,  (),  U,  S,  .  . .  will  produce 
the  motion  of  the  two  systems  connected  at  the  points  p,  g,  r,  s,  .  .  . 
Now  the  same  forces  cancel  each  other  out  when  applied  to  the  system 
of  levers  alone.  The  reunion  of  the  two  systems  could  not  perturb 
this  equilibrium.  Whence  it  is  impossible  that  the  forces  P,  Q9  jR,  S,  .  .  . 
should  produce  the  movement  of  the  given  system.  This  is  true 
for  any  other  displacement  for  which  the  total  moment  of  the  forces 
is  zero.  "  And  from  this  can  be  deduced  the  following  particular 
conclusion,  which  includes  the  principle  of  virtual  velocities.  If, 
of  all  the  possible  displacements,  there  is  none  which  corresponds 
to  a  zero  moment,  there  must  be  equilibrium.  " 

Moreover  it  suffices  that  the  sum  of  the  moments  should  not  be  negative. 
Indeed,  "  it  is  easily  proved,  by  the  theory  of  the  lever  alone,  that 
these  forces  applied  [to  the  levers  alone]  cannot  produce  a  displacement 
for  which  the  total  moment  is  positive.  And  since  it  is  supposed 
that  the  presence  of  obstacles  makes  all  other  displacements  impossible, 
it  is  necessary  that  when  the  forces  act  on  the  levers,  they  maintain 
them  in  equilibrium.  This  will  still  be  true  if  the  first  system  is  applied 
to  the  second.  Therefore  these  forces  cannot  separately  produce 
the  displacement  in  question  in  the  first  system.  For  this  displacement 
would  also  accur  if  the  second  system  were  applied  to  the  first,  and 
we  have  just  seen  that  it  is  then  impossible. 

"  Conversely,  if  some  powers  maintain  any  material  system  in 
equilibrium,  there  can  be  no  displacement  of  the  system  possible 
for  which  the  sum  of  the  moments  can  be  negative.  This  is  proved 
in  the  following  way.  If  it  is  assumed  that  the  system  can  move 
into  such  a  position  that  the  moment  of  the  forces  is  negative,  it  must 
be  concluded  that  equilibrium  does  not  exist.  For  the  equilibrium 
would  not  cease  to  exist  if  this  displacement  became  the  only  possible 
one.  It  is  easy  to  represent  this  last  effect  by  imagining  assemblies 
of  levers,  similar  to  those  which  have  been  described  above,  between 
all  the  points  p,  5,  r,  s,  ...  of  the  system,  and  capable  of  the  virtual 
velocities  which  correspond  to  the  displacement  concerned.  It  is 
unnecessary  to  show  that  the  equilibrium  will  not  be  disturbed  by  the 
addition  of  these  levers.  Now  it  is  impossible  that  there  should  not 
be  motion,  because  the  forces  will  find  themselves  applied  to  an  as- 


366  THE   EVOLUTION   OF   CLASSICAL 

sembly  of  levers  which  could  not  fail  to  be  displaced  if  the  sum  of  the 
moments  of  the  forces  were  negative,  just  as  it  follows  from  the  theory 
of  the  lever.  Therefore  it  is  necessary  that  the  sum  of  the  moments 
of  the  forces  should  never  be  negative.  " 

Finally  Fourier  introduced  the  distinction  between  bilateral  and 
unilateral  constraints. 

"  Whenever  the  displacements  of  which  the  body  is  capable  are 
determined  by  the  equations  of  condition  which  they  must  satisfy, 
the  total  moment  of  the  forces  cannot  be  positive  when  the  forces 
are  in  equilibrium.  For  if  this  moment  were  positive,  the  moment 
corresponding  to  the  contrary  displacement  would  be  negative.  Now 
as  this  latter  displacement  is  equally  possible,  since  it  satisfies  the 
equations  of  condition,  the  forces  could  not  cancel  each  other  out.  .  .  . 
That  is  why  it  is  necessary,  in  this  case,  that  the  sum  of  the  moments 
of  the  forces  must  be  zero  in  order  that  there  should  be  equilibrium. 
This  is  the  true  meaning  of  the  principle  of  virtual  velocities.  But 
if  the  displacements  are  not  prohibited  by  the  equations  of  condition 
— which  often  happens — the  equilibrium  can  subsist  without  the 
moment  of  the  forces  being  zero,  provided  that  it  is  not  negative.  " 

In  connection  with  this  demonstration,  Jouguet  has  remarked 
that  Fourier  thus  established  the  principle  of  the  equivalence  of  cons 
traints,  which  may  be  stated  in  the  following  way. 

"  Let  there  be  a  system  of  points  acted  upon  by  forces  F  and  bound 
by  constraints  (L).  Replace  the  constraints  (L)  by  the  constraints 
(I/)  which  preserve  the  same  elementary  mobility  as  the  constraints 
(L).  In  order  that  the  system  should  be  in  equilibrium,  it  is  sufficient 
and  it  is  necessary  that  the  forces  F  should  be  in  equilibrium  under 
the  constraints  (L7).  "  1 

Fourier's  analysis  breaks  down  if  the  constraints  (L)  and  (I/) 
introduce  resistances  to  motion,  even  if  (I/)  and  (L')  assure  the  same 
kinematic  mobility  of  the  system.  "It  is  not  only  the  kinematic 
mobility  which  must  be  preserved  but  also,  as  it  were,  the  dynamical 
mobility.  "  2 

1  L.  AT.,  Vol.  II,  p.  171. 

2  Ibid.,  p.  172. 


CHAPTER   THREE 


THE  PRINCIPLE  OF  LEAST  CONSTRAINT  (1829) 


The  principle  of  least  constraint  was  stated  by  Gauss  in  a  paper 
called  Vber  ein  neues  Grundgesetz  der  Mechanik  in  Volume  IV  of  the 
Journal  de  Crelle  (1829).1 

He  wrote,  "  It  is  known  that  the  principle  of  virtual  velocities 
makes  the  whole  of  statics  a  matter  of  analysis,  and  that  d'Alembert's 
principle,  in  its  turn,  reduces  dynamics  to  statics.  In  the  nature 
of  things,  there  can  exist  no  new  principle  in  the  science  of  equilibrium 
and  motion  which  is  not  included  in  the  two  preceding  principles, 
or  which  cannot  be  deduced  from  them. 

44  However,  such  a  principle  may  not  be  without  value.  It  is 
always  interesting  and  instructive  to  regard  the  laws  of  nature 
from  a  new  and  advantageous  point  of  view,  so  as  to  solve 
this  or  that  problem  more  simply,  or  to  obtain  a  more  precise  pre 
sentation. 

44  The  great  geometer  [Lagrange]  who  succeeded  so  brilliantly 
in  constructing  mechanics  from  the  principle  of  virtual  velocities, 
had  no  disdain  to  generalise  and  to  develop  the  principle  of  least  action 
in  Maupertuis'  sense,  but  rather,  he  used  it  to  great  advantage.  '* 

To  Gauss,  the  principle  of  virtual  velocities  was  the  prototype 
of  the  principles  of  mechanics.  But  this  principle  was  not  intuitive, 
and  demanded  a  special  treatment  in  order  that  it  might  be  extended 
from  statics  to  dynamics.  That  is  why  Gauss  believed  it  useful  to 
state,  in  the  following  form,  a  new  principle. 

"  The  motion  of  a  system  of  particles  connected  together  in  any 
way,  and  whose  motions  are  subject  to  arbitrary  external  restrictions, 
always  takes  place  in  the  most  complete  agreement  possible  with 
the  free  motion  (in  moglich  grosster  ffbereinstimmung  mil  der  freien 
Bewegung)  or  under  the  weakest  possible  constraint  (unter  moglich 
kleinstem  Zwange).  The  measure  of  the  constraint  applied  to  the 

1  The  complete  Works  of  Gauss  (french  edition),  Vol.  V,  p.  25. 


368  THE   EVOLUTION    OF    CLASSICAL 

system  at  each  elementary  interval  of  time  is  the  sum  of  the  products 
of  the  mass  of  each  particle  with  the  square  of  its  departure  from 
the  free  motion.  " 

Let  m,  m',  m"  .  .  . 

be  the  masses  of  the  points  of  the  system  and 

a7  a  ,  a    ... 

be  their  positions  at  the  time  t.     Let 

6,  b\  b"  ... 

be  the  positions  which  they  would  assume  at  the  time  t  -j-  dt  under  the 
action  of  the  forces  which  are  applied  to  them,  and  because  of  their 
velocities  at  the  time  J,  if  it  were  supposed  that  they  were  completely 
free  of  all  constraint.  The  actual  positions,  c,  c',  c",  ...  of  the  different 
points  will  be  such  that,  while  being  compatible  with  the  constraints, 
they  minimise  the  sum 


The  equilibrium  is  evidently  a  special  case  of  the  general  law  according 
to  which 

m(a&)2  +  ro'(a'&')2  +  ™'V6")2  +  •  •  - 
is  a  minimum. 

Gauss  wrote,  "  This  is  how  our  principle  can  be  deduced  from 
principles  already  known.  The  force  which  is  exerted  on  the  particle  m 
is  evidently  composed  of  two  ;  first,  the  force  that,  taking  account 
of  the  velocity  at  the  time  J,  brings  the  particle  from  a  to  c  in  the  time 
dt  ;  secondly,  the  force  which,  in  the  same  time,  would  bring  the  same 
element  from  c  to  6,  if  it  were  supposed  to  be  free  and  to  start  from 
rest.  [This  is  the  same  decomposition  as  that  which  d'Alembert  used.] 
Similarly  for  all  particles. 

"  By  d'Alembert's  principle  the  points  m,  m\  m"  must  be  in  equi 
librium,  because  of  the  constraints  of  the  system,  under  the  sole  action 
of  the  second  forces  acting  along  c6,  c'6',  c"6",  .  .  .  According  to 
the  principle  of  virtual  velocities  this  equilibrium  requires  that  [the 
sum  of  the  virtual  works]  should  be  zero  for  every  virtual  displacement 
which  is  compatible  with  the  restraints.  Or,  more  accurately,  this 
sum  should  never  be  positive. 

"  Then  let  y,  y',  y",  ...  be  different  positions  of  c,  c',  c",  .  .  .  which 
are  compatible  with  the  constraints.  Let  0,  0',  6",  ...  be  the  angles 
that  cy,  c'y',  c"y",  .  .  .  make  with  c6,  c'b',  c"6",  .  .  .  Then 

£mc&  •  cy  •  cos  0 
will  be  zero  or  negative. 


THE    PRINCIPLE    OF   LEAST    CONSTRAINT  369 

"  Since 

yb*  =  cb2  -f-  cy2  —  2ci  -  cy  •  cos  6 

it  is  clear  that 

y>^62  =  Vm7&2  +  £m~^>  —  2  £mc6  -  cy  -  cos  0. 
Therefore 


will  always  be  positive.     Therefore,  finally, 


will  always  be  a  miniraum.     Q.  E.  D.  " 

In  conclusion,  Gauss  emphasised  the  fact  that  free  motions,  when 
they  are  incompatible  with  the  constraints,  are  modified  in  Nature 
in  the  same  way  that  experimental  data  are  modified,  by  the  method 
of  least  squares,  so  as  to  be  compatible  with  a  necessary  relation  bet 
ween  the  measured  quantities. 


24 


CHAPTER   FOUR 


RELATIVE  MOTION 
RETURN  TO  A  PRINCIPLE  OF  CLAIRAUT 

CORIOLIS'  THEOREMS 
FOUCAULT'S  EXPERIMENTS 


1.  RETURN  TO  A  PRINCIPLE  OF  CLAIRAUT  (1742). 

The  first  outline  of  a  theory  of  relative  motion  appeared,  as  we 
have  seen,  in  Huyghens'  De  vi  centrifugal 

Though  he  did  not  resolve  all  the  difficulties  of  relative  motion, 
Clairaut  had  the  indisputable  merit  of  generalising  Huyghens'  concep 
tions.  This  he  did  in  a  paper  called  Sur  quelques  principes  qui  donnent 
la  solution  d*un  grand  nombre  de  problemes  2  which  could  not  have 
escaped  the  attention  of  Coriolis. 

Clairaut  set  out  to  find  "  what  happens  to  any  system  of  bodies, 
actuated  by  gravity  or  other  accelerating  forces,  when  this  system 
is  attached  by  some  part  to  a  plane  and  is  carried  with  this  plane 
in  some  curvilinear  motion.  "  He  introduced  "  the  general  principle 
for  finding  the  motions  of  systems  of  bodies  carried  along  by  the  planes 
on  which  they  are  placed  "  in  the  following  way. 

"  Imagine  that  the  rectangle  FGHI  is  placed  between  two  curves 
AB  and  CD  and  that,  when  the  corner  G  is  moved  at  will  on  the  curve 
AB^  the  corner  I  follows  the  curve  CD. 

"  Suppose  now  that  one  of  the  bodies  M  of  the  system  given, 
accelerated  by  gravity  or  by  any  other  forces,  describes  the  curve 
MJU,  because  of  the  properties  of  the  system. 

"  We  seek  the  accelerating  or  retarding  force  that  the  motion  of 
the  plane  FGHI  gives  the  body  M.  We  shall  start  by  tracing  the  curve 
PQ  that  the  point  M  would  describe,  during  the  motion  of  FGHI9 
if  it  were  fixed  in  the  plane  FGHI.  We  shall  then  determine  the  velo- 

1  See  above,  Part  II,  p.  194. 

2  Memoires  de  V Academic  des  Sciences,  1742,  p.  1. 


RELATIVE   MOTION 


371 


city  with  which  it  would  move  along  this  line,  which  will  only  depend 
on  the  given  velocity  of  G  and  the  curves  AB  and  CD.  This  done, 
we  shall  seek  the  accelerating  forces  which  it  is  necessary  to  suppose 
distributed  in  the  space  AB,  CD  in  order  that  the  body  M,  left  to 
itself  with  the  velocity  that  it  has  at  M  on  Mm,  might  travel  the  line 
PQ  Let  MS,  for  example,  represent  what  this  force  would  be  at  M. 
I  say  that  by  producing  MS  and  taking  MT  =  MS,  the  straight 
line  MT  represents  the  force  by  which  the  motion  of  the  plane  FGHI 
alters  the  velocity  of  M  on 


H 


f" 


"  To  prove  this,  I  start  by  distinguishing  the  particle  M  from  the 
fixed  point  of  FGHI  which  corresponds  to  it,  and  I  call  this  fixed  point 
M'.  I  then  remark  that  if,  at  the  instant  that  the  body  M  has  travelled 
along  M/*  and  the  body  M'  has  traveUed  along  M'm,  the  curves  AB 
and  CD  were  suddenly  removed  and  the  plane  FGHI  were  allowed 
to  move  uniformly  with  the  velocity  of  M'  along  M'm,  the  system 
which  is  on  the  plane  FGHI  would  necessarily  move  in  the  same  way 
as  if  this  plane  were  fixed.  I  add  to  this  remark  that  the  reason  why 
the  motion  along  the  arc  Mm  is  altered  in  the  curvilinear  motion 
of  the  plane  FGHI  is  that,  in  order  to  produce  the  curvilinear  motion, 
it  is  necessary  to  imagine  that  the  body  M'  receives  an  impulse  MS  at 
the  instant  that  it  has  traveUed  along  M'm,  and  that  the  body  M  does 


372  THE   EVOLUTION   OF   CLASSICAL 

not  receive  this  impulse.  For  if  the  body  M  received  this  impulse, 
the  motion  of  the  system  would  be  exactly  the  same  as  if  the  plane 
FGHI  were  fixed.  Given  this,  I  say  that  it  is  the  same  whether  M' 
receives  an  impulse  and  M  does  not,  or  whether  M  receives  it  in  the 
opposite  direction  and  M'  does  not. 

"  Therefore  the  plane  FGHI  can  be  regarded  as  fixed  and  it  can 
be  supposed  that  the  body  experiences  the  action  of  the  given  forces 
as  well  as  the  action  of  the  forces  MT.  " 

In  short,  in  this  way  Clairaut  arrives  at  an  estimate  of  the  quantity 
myr  in  the  relative  motion.  This  estimate  is  Fa  —  mye,  where  ye 
is  the  dragging  acceleration. 

We  know  that  this  principle  is  incomplete.  However,  it  led  Clairaut 
to  correct  results  when  he  confined  himself  to  applying  the  principle 
of  kinetic  energy  to  relative  motion.  For  it  is  known  that  Coriolis' 
complementary  force  of  inertia  does  no  work. 

Incomplete  though  it  may  be,  Clairaut's  argument  has  a  synthetic 
value  and,  for  this  reason,  it  is  of  some  use  to  complete  it.  This, 
in  fact  was  accomplished  by  Joseph  Bertrand  in  1848.1 

We  shall  not  follow  Bertrand's  analysis  here,  but  shall  present 
an  alternative  method  by  which  the  argument  may  be  completed. 

With  respect  to  a  fixed  reference  system  in  which  the  law  mya  =  Fa 
is  valid,  the  particle  M  describes  the  curve  MMX  between  the  times  t 
and  t  +  dt. 

Under  the  same  conditions  its  coincident  point  M'  is  connected 
to  a  moving  reference  system  (S)  which  has  any  arbitrary  continuous 
motion  relative  to  an  absolute  reference  system.  It  then  describes 
the  curve  M'M{  between  the  times  t  and  t  -f-  dt.  If  the  particle  M 
had  retained  the  absolute  velocity  va  which  it  had  at  the  time  £,  it 
would  have  travelled  to  M2  in  the  time  dt,  where  MM2  =  vadt.  Simi 
larly,  if  M'  had  retained  its  absolute  velocity  t?e,  it  would  have  travelled 

1  Journal  de  VScole  poly  technique,  Vol.  32,  p.  148,  1848. 

In  this  connection,  J.  BERTRAND  brings  grist  to  the  mill  of  history.  "  Too  often, 
after  having  studied  analytical  mechanics,  a  man  helieves  it  is  useless  to  seek  to  com 
plete  this  study  by  the  reading  of  the  scattered  works  with  which  the  predecessors  of 
LAGRANGE  enriched  the  academic  collections  of  the  XVIIIth  Century.  I  believe  that 
this  tendency,  unfortunately  very  common,  is  such  as  to  destroy  the  progress  of  me 
chanics,  and  that  it  has  already  produced  unfortunate  results.  The  too  common  custom 
of  deducing  formulae  has,  to  some  extent,  led  to  the  loss  of  a  proper  respect  for  the 
truths  of  mechanics  considered  in  themselves. " 

In  BERTRAND'S  opinion,  "  M.  Coriolis,  without  knowing  it,  has  done  the  same  as 
the  illustrious  Clairaut.  " 

That  is  his  opinion.  It  is  true  that  CORIOLIS  does  not  refer  to  CLAIRAUT.  It  is 
also  true  that  he  went  further  than  him.  But  it  seems  unlikely  to  us  that  CLAIRAUT'S 
paper  could  have  been  omitted  from  CORIOLIS'  reading. 


RELATIVE   MOTION  373 

to  MS,  where  M'M^  =  vedt.  Follow  on  let  ~vr  be  the  relative  velocity 
of  M  in  the  system  (S)  at  the  time  t.  In  the  triangle  MM2M^  using 
the  composition  of  velocities  (va  =  ve  +  vr),  it  is  seen  that  M^M2  =  vrdt. 
Now  give  the  reference  system  (S)  and  the  particle  M  the  same 
absolute  motion  defined  in  the  following  way  —  first,  a  translation 

jVf'Mg  =  _  -  ye<fr2  which  annuls  the  deviation  M%M(  of  the  point  M'; 

secondly,  the  rotation  —  wdt  at  Mg,  to  annul  the  effect  of  the  absolute 
rotation  of  the  system  (S),  considered  as  a  solid,  between  the  times  t 
and  t  +  dt.  Thus,  taking  account  of  the  motion  which  it  originally 
possessed,  the  system  (S)  will  have  experienced  a  rectilinear  translation 
M'Mg  of  velocity  ve. 


M2 


Mjft+ctr) 
Fig.  109 

Correspondingly,  the  particle  M  experiences  the  translation  —  ^ 
and,  to  the  third  order,  a  displacement 

—  vdt  A  MiM2  -  — (S  A  vr)dt*. 

These  two  displacements  of  the  particle  M  in  absolute  space  can 
be  fictitiously  imputed  (as  Clairaut  realised)  to  forces  — mye  and 

—  2m  (co  A  ^r)  respectively. 

Therefore,  between  t  and  t  +  dt,  the  fundamental  law  of  can  be 
written  in  the  system  (S),  corrected  in  its  motion  in  this  way  (that  is, 
in  rectilinear  and  uniform  translation  with  velocity  vc),  in  the  form 

myr  ==  Fa  —  (mye  +  2m  (CD  A  vr)}. 
This  completes  and  rectifies  Clairaut's  principle. 


374  THE  EVOLUTION  OF  CLASSICAL 

2.  CORIOLIS'  FIRST  THEOREM. 

CoriohV  name  has  remained  associated  with  the  law  of  the  com 
position  of  accelerations.  This  law  belongs  to  the  domain  of  pure 
kinematics —  that  is  the  way  it  is  taught  at  present,  before  its  dynamical 
consecpiences  are  explored. 

Historically,  Coriolis  was  concerned  with  the  theory  of  water- 
wheels  when  he  embarked  on  his  study  of  relative  motion.  This 
theory  had  already  been  studied  by  Jean  Bernoulli,  Euler,  Borda, 
Navier  and  Ampere.  To  progress  beyond  the  earlier  work,  it  was 
necessary  to  study  the  following  general  problem. 

"  To  find  the  motion  of  any  machine  in  which  certain  parts  are  moved 
in  a  given  way.  " 

Here  we  shall  follow  Coriolis'  first  paper,  which  was  read  to  the 
Academie  des  Sciences  on  June  6th,  1831,  and  was  printed  in  the  Journal 
de  r£cole  polytechnique.1 

Coriolis  considers  two  reference  frames.  One,  Ox^y^z^  is  fixed 
—absolute.  The  other,  OXYZ,  is  movable— relative.  Let  f,  77,  £ 
be  the  absolute  coordinates  of  the  origin  of  the  movable  frame  and 
(a,  a',  a"),  (ft,  b\  V'},  (c,  c',  c")  be  the  direction  cosines  of  the  mov 
able  axes  with  the  fixed  axes. 

The  constraints  which  exist  during  the  motion  are  supposed  to 
be  perfect  and  expressible  in  finite  terms  in  the  relative  coordinates. 
Let  £  be  the  force  of  constraint  applied  at  a  point  of  the  system. 
Using  the  method  of  Lagrange  multipliers,  Coriolis  writes  the  projections 
of  the  force  on  the  movable  axes  as 

dL       m 

'  dx       ^  dx 

2    'iL       m 


Returning  to  the  fixed  axes,  Coriolis  calculates  the  total  force 
— the  given  force  and  the  force  of  constraint — acting  on  one  point  of 
the  system.  Briefly,  this  may  be  written 

(A)  mya  =  P  +  £ 

Coriolis  sums  the  equations  (A)  after  multiplying  them  by  the 
relative  displacement  dsr.  Under  these  conditions  the  forces  due  to 

1  21st  caHer,  1832,  p.  268. 


RELATIVE    MOTION  375 

the  constraints  vanish.  But  in  order  to  perform  a  particular  calculation, 
it  is  necessary  to  express  the  absolute  accelerations  ya  as  functions 
of  the  relative  coordinates  and  velocities  as  well  as  the  dragging  motion. 

For  this  purpose  Coriolis  distinguishes  total  differentials  of  the 
true  motion,  indicated  by  the  symbol  d,  from  the  differentials  obtained 
by  varying  only  the  quantities  (a,  6,  ...  cff).  This  differentiation 
is  indicated  by  the  symbol  d^  and  corresponds  to  a  variation  of  the 
orientation  of  the  movable  system  in  which  the  quantities  x,  j,  s, 
$  ,  77,  C  remain  constant. 

46  If  the  points  are  not  displaced  relatively  to  the  moving  axes, 
they  only  have  the  dragging  motion  l  corresponding  to  these  axes, 
whose  origin  is  supposed  immovable  and  which  only  have  a  rotational 
motion  about  this  origin.  " 

Thus,  "  by  omitting  to  write  the  denominators  dt2  under  the  dif 
ferentials,"  Coriolis  writes  the  components  of  the  absolute  acceleration 
in  the  form 


+  cd*z  +  2dxda  +  2dydb  +  2dzdc 
(B)     d*yi  =  ... 


Terms  of  the  equation  (A)  such  as 


m 


then  appear  to  him  as  "  the  components,  with  respect  to  the  fixed  axes, 
of  the  forces  Fe  which  would  produce  the  motion  which  each  point  would 
take  if  it  remained  in  the  same  place  with  respect  to  the  moving  axes.  " 
We  have  quoted  from  the  paper  of  Coriolis  in  order  to  illustrate  the 
development  of  his  thought,  how  he  did  not  pause  on  the  kinematic 
aspect  of  the  problem,  but  went  directly  to  the  cause  that  would  be 
able  to  produce  the  dragging  motion  (in  the  modern  sense  now). 

Then  Coriolis  returns  to  the  moving  axes  by  first  substituting 
the  equations  (B)  in  the  equations  (A),  then  by  projecting  on  the 
moving  axes.  He  obtains  the  relation 

-  2mdy(adb  +  o!dV  +  a"db")  +  2mdz(adc  +  a'dc'  + 

_  .  „  dL  .     dM 


+ ... 

1  This  definition  does  not  coincide  with  that  which  is  now  common. 


376  THE   EVOLUTION   OF    CLASSICAL 

By  summing  for  all  the  points  of  the  system,   after  multiplying 
by  the  relative  displacements  dx,  dy,  dz,  he  arrives  at 


Ydy  +  Zdz). 

The  cross-terms  in  dx,  dy,  dz,  vanish  in  the  summation  because  of  the 
relations  between  the  direction  cosines. 

If  VT  is  the  relative  velocity  of  any  point  of  the  system  with  respect 
to  the  moving  axes,  and  Pe  the  force  which  is  e<jual  and  opposite 
to  Fe,  P  the  given  force,  Coriolis  writes 


or,  by  integration, 


"  Thus  the  principle  of  living  forces  is  still  true  for  motion  relative 
to  moving  axes,  provided  that  there  are  added  to  the  quantities  of  action 
[that  is,  of  work]  J  Pdsr  cos  (P-dsr)  calculated  from  the  given  forces 
and  the  arcs  dsr  described  in  the  relative  motion,  other  quantities  of  action 
which  are  due  to  the  forces  Pe.  These  forces  are  supposed  equal  and 
opposite  to  those  which  it  would  be  necessary  to  apply  to  each  moving 
point  in  order  to  make  it  take  the  motion  that  it  would  have  had  if  it  were 
in  invariably  connected  to  the  moving  axes.  " 

This  is  Coriolis'  first  theorem,  which  essentially  belongs  to  the 
dynamics  of  relative  motion.  Coriolis  applied  it  to  the  "  quantity 
of  action  "  transmitted  to  the  machine  which  carried  the  movable 
axes.  We  shall  not  follow  him  in  this  application,  where  he  made 
simultaneous  use  of  the  theorem  of  kinetic  energy  in  the  absolute 
motion  and  the  relative  motion. 

Coriolis  remarked  that  when  the  question  was  that  of  the  equilibrium 
of  a  fluid  contained  in  a  vessel  turning  about  an  axis,  "  it  is  immediately 
seen  that  it  is  necessary  to  introduce  actions  equal  to  the  centrifugal 
forces.  But  it  is  not  the  same  for  the  principle  of  living  forces  applied 
to  the  relative  motion.  It  would  be  mistaken  to  regard  this  proposition 
as  evident  ;  to  proceed  in  this  way  for  any  other  equation  than  that 
of  the  living  forces  would  be  to  arrive  at  false  results.  " 

In  conclusion,  Coriolis  declared,  "  Such  are  the  principal  results  of 
this  paper.  It  seems  that  the  principle  from  which  they  stem  may 
find  many  applications  in  the  theory  of  machines,  provided  that  it  is 
supplemented  by  a  number  of  propositions  from  rational  mechanics.  " 


RELATIVE   MOTION  377 

3.    CORIOLIS*    SECOND    THEOREM. 

Coriolis  expectation  was  more  than  fulfilled.  For  his  first  paper 
already  contained  the  germ  of  the  fundamental  theorem  with  which 
his  name  is  now  associated.  This  vital  point  is  his  equation  (C),  which 
Coriolis  did  not  himself  analyse  thoroughly,  anxious  as  he  was  to  cal 
culate  the  quantity  of  action  transmitted  to  his  water-wheels. 

In  a  second  paper,  Sur  les  equations  du  mouvement  relatifdes  systemes 
de  corps,1  Coriolis  wrote — 

"  In  this  paper  I  give  the  following  general  proposition — that  to 
establish  an  equation  of  the  relative  motion  of  a  system  of  bodies 
or  of  any  machine,  it  suffices  to  add  to  the  existing  forces  two  kinds 
of  supplementary  forces.  The  first  are  always  those  to  which  it  is 
necessary  to  have  regard  for  the  equation  of  living  forces  ;  that  is, 
which  are  the  forces  opposed  to  those  which  are  able  to  keep  the  part 
icles  constantly  connected  with  movable  planes.  The  second  are 
directed  perpendicularly  to  the  relative  velocities  and  to  the  axes 
of  rotation  of  the  movable  planes  ;  they  are  equal  to  twice  the  product 
of  the  angular  velocity  of  the  movable  planes  and  the  relative  quantity 
of  motion  on  a  plane  perpendicular  to  this  axis. 

"  The  latter  forces  are  most  closely  analogous  to  ordinary  centrifugal 
forces.  To  display  this  analogy  it  suffices  to  remark  that  the  centri 
fugal  force  is  equal  to  the  quantity  of  motion  multiplied  by  the  angular 
velocity  of  the  tangent  to  the  curve  described  ;  that  it  is  directed 
perpendicularly  to  the  velocity  and  in  the  osculatory  plane,  this  is, 
perpendicularly  to  the  axis  of  rotation  of  the  tangent.  Thus  in  order 
to  pass  from  ordinary  centrifugal  forces  to  the  second  forces  which 
occur,  multiplied  by  two,  in  the  preceding  statement,  it  is  only  necessary 
to  replace  the  angular  velocity  of  the  tangent  by  that  of  the  movable 
planes,  and  to  substitute  for  the  direction  of  the  axis  of  rotation  of 
this  tangent,  the  direction  of  the  axis  of  rotation  of  the  same  movable 
planes.  In  other  words,  it  suffices  to  substitute  everything  which 
is  related  in  magnitude  and  direction  to  the  rotation  of  the  tangent 
by  what  is  related  to  the  rotation  of  the  movable  planes,  and  to 
multiply  the  forces  thus  obtained  by  two. 

"  It  is  because  of  this  analogy  that  I  concluded  that  these  forces 
must  be  named  compound  centrifugal  forces.  Indeed,  they  have 
some  of  the  characteristics  of  the  relative  motion  because  of  the  quantity 
of  motion  and  some  of  the  characteristics  of  the  motion  of  the  movable 
planes  through  the  use  of  their  axes  of  rotation  and  angular  velocity. 

1  Journal  de  r£cole  polytechnique,  24th  cahier,  1835,  p.  142, 


378  THE   EVOLUTION   OF    CLASSICAL 

"  Therefore  it  will  be  said  that,  for  an  equation  of  relative  motion 
which  is  not  that  of  the  living  forces,  it  is  necessary  to  introduce  twice 
the  compound  centrifugal  force. 

"  The  theorem  which  I  presented  at  the  Academic  des  Sciences 
in  1831  consists  of  the  disappearance  of  the  compound  centrifugal  forces 
from  the  equation  of  the  living  forces.  It  now  becomes  a  particular 
case  of  the  more  general  statement  on  the  introduction  of  these  com 
pound  centrifugal  forces.  " 

Coriolis9  demonstration  depends  directly  on  the  equation  (C)  al 
ready  written  above.  Indeed,  if  p,  q,  r  are  the  "  three  angular  velocities 
of  the  movable  planes  about  their  axes,  "  (C)  can  be  written  in  the 
form 


dzx  I  d  dz\  9L         dM 


(C) 


"  The  terms  in  p,  q,  r,  dx,  dy  and  dz  in  the  above  equation  are 
twice  the  components,  along  the  moving  axes,  of  a  force  directed 
perpendicularly  to  the  plane  of  the  axis  of  rotation  and  the  relative 
velocity.  The  magnitude  of  this  force  will  be  the  product  of  the 
angular  velocity  Y^  p2  +  g2  +  r2  with  the  projection,  on  a  plane 
perpendicular  to  the  axis  of  rotation,  of  the  quantity  of  motion 
due  to  the  relative  velocity  of  the  particle.  The  sense  in  which 
this  force  will  be  carried,  with  respect  to  a  motion  which  carries 
the  axis  of  rotation  towards  the  relative  velocity,  will  be  the 
same  as  that  of  the  axis  of  rotation  with  respect  to  the  velocity  of 
rotation.  " 

"  The  expressions  for  the  forces  which  must  be  added  to  the  given 
forces  in  order  to  obtain  the  expressions  for  the  forces  in  the  relative  motions 
are —  first,  those  which  are  opposed  to  the  forces  able  to  produce,  for  each 
particle,  the  motion  which  it  would  have  if  it  were  connected  to  movable 
planes ;  secondly,  twice  the  compound  centrifugal  forces.  " 

This  is  valid  for  a  particle  of  the  system.  Coriolis  then  considers 
the  virtual  velocities — the  displacements  dx,  dy,  dz  in  the  relative  mo 
tion — compatible  with  the  relative  constraints 

L  =  0         M  ==  0     etc. 

These  relative  constraints,  supposed  to  be  perfect,  will  disappear 
on  combining  the  equations  (C),  giving 


RELATIVE   MOTION  379 

.dyds  -  dzdy\ 


If  ds  is  the  actual  virtual  displacement  and  ds  the  virtual  relative 
displacement  ;  if  a,  ft  and  y  are  the  direction  cosines  of  the  instanta 
neous  rotation  oj(p,  q,  r)  with  respect  to  the  moving  axes  and  A,  u,  v 
are  the  direction  cosines  of  the  normal  to  the  plane  (ds,  <5s),  the  Coriolis' 
complementary  term  becomes 

2o>V  m  —  (5s  sin  (rfs,  ds)  (od  +  $"  +  yv). 

Therefore,  "  In  order  to  obtain  an  equation  of  the  relative  motion 
it  is  necessary  to  add  to  the  terms  ordinarily  existing  for  absolute  motion 
—  first,  those  which  arise  from  the  forces  which  are  able  to  force  the  particles 
to  remain  connected  to  the  movable  planes  ;  and,  in  addition,  a  term 
which  is  equal  to  twice  the  velocity  of  rotation  multiplied  by  the  algebraic 
sum  of  the  projections,  on  a  plane  perpendicular  to  the  axis  of  rotation 
of  these  planes,  of  all  the  areas  of  the  parallelograms  defined  by  the  effective 
quantities  of  motion  and  the  virtual  velocities.  " 

For  the  equation  of  the  kinetic  energy,  each  area  is  zero.  For 
the  virtual  velocity  coincides  with  the  effective  velocity  (or  rather, 
with  the  true  displacement).  Thus  Coriolis'  two  theorems  are  linked 
together. 

We  have  said  enough  to  illustrate  the  development  of  Coriolis' 
thought.  In  fact,  he  complicated  his  task  by  isolating  the  law  of 
the  composition  of  the  accelerations  —  singularly  hidden  —  from  the 
already  difficult  problem  of  the  dynamics  of  systems.  It  is  rather 
interesting  to  remark  in  passing  that  Coriolis  composed  two  acceler 
ations  by  summing 


,    ,  _ 

at*     dp       ' 

Very  fortunately,  however,  this  procedure  entailed  no  risk  because 
it  reduced  to  connecting  together  two  terms  of  the  unique  dragging 
acceleration  ye  (in  the  modern  sense). 

We  stress  the  fact  that  Coriolis  did  not  deal,  in  fact,  with  kinematics. 
He  argued  exclusively  from  a  dynamical  point  of  view,  using  forces, 


380  THE    EVOLUTION    OF    CLASSICAL 

and  only  endowed  products  such  as  mye  with  a  physical  significance. 
His  aim  was  to  find  an  equation  of  the  relative  motion  which  might 
be  independent  of  the  constraints,  supposed  to  be  holonomic  and 
perfect.  It  is  for  this  reason  that  he  first  encountered  the  theorem 
of  the  kinetic  energy,  in  which  the  compound  centrifugal  force 
vanished.  Then  he  was  able  to  give  a  more  general  equation  in  which 
the  complementary  term  appeared.  All  this  only  makes  his  discovery 
more  remarkable. 

4.  THE  EXPERIMENTS  OF  FoucAULT  (1819-1868). 

In  the  strict  sense,  mechanics  which  is  referred  to  terrestrial  axes 
should  take  account  of  CoriohV  compound  centrifugal  force.  Never 
theless,  we  have  already  had  occasion  to  remark  l  that  the  deviation 
of  heavy  bodies  towards  the  East  can  be  predicted  by  a  very  simple 
intuitive  argument.  Moreover,  as  early  as  1833,  Reich  studied  free 
fall  in  a  mine- shaft  at  Freiberg  (Saxony).  The  depth  of  the  mine 
was  188  m.,  and  he  observed  an  average  deviation  of  28  millimetres 
in  106  separate  observations. 

In  1851  Foucault  published  a  paper  called  Demonstration  physique 
du  mouvement  de  rotation  de  la  Terre  au  moyen  du  pendule.2 

This  demonstration  made  no  appeal  to  Coriolis'  work — only  after 
the  event  did  occur  a  mathematical  literature.  Foucault,  who  had  been 
a  mediocre  pupil  at  school,  was  a  natural  physicist  and  an  incomparable 
experimenter.  However,  he  started  work  as  the  scientific  member 
of  the  staff  of  the  Journal  des  Debats. 

He  set  out  to  experiment  on  the  direction  of  the  plane  of  oscillation 
of  a  pendulum.  If  the  observer  is  at  first  supposed  at  the  pole  (North 
or  South)  and  the  pendulum  is  reduced  to  a  homogeneous  spherical 
mass  suspended  from  an  absolutely  fixed  point,  then  if  this  point  is 
exactly  on  the  axis  of  rotation  of  the  Earth,  the  plane  of  oscillation 
remains  fixed  in  space.  "  The  motion  of  the  Earth,  which  forever 
rotates  from  west  to  east,  will  become  appreciable  in  contrast  with 
the  fixity  of  the  plane  of  oscillation,  whose  trace  on  the  ground  will 
seem  to  be  actuated  by  a  motion  conforming  to  the  apparent  motion 
of  the  celestial  sphere.  And  if  the  oscillations  can  continue  for  twenty- 
four  hours,  in  this  time  the  plane  will  execute  a  whole  revolution  about 
the  vertical  through  the  point  of  suspension.  " 

But  Foucault  also  remarked  that  in  reality  it  is  necessary  to  "  take 
support  on  moving  earth ;  the  rigid  pieces  to  which  the  thread  of  the 

1  See  above,  Part  I,  p.  63. 

2  Comptes  rendus  de  I' 'Academic  des  Sciences,  Vol.  32,  p.  135  (February  3rd,  1851). 


RELATIVE   MOTION  381 

pendulum  is  attached  cannot  be  isolated  from  the  diurnal  motion. 
It  should  be  borne  in  mind  that  this  motion,  communicated  to  the 
thread  and  the  mass  of  the  pendulum,  might  alter  the  direction  of 
the  plane  of  oscillation.  "  Nevertheless,  experiment  shows  that 
"provided  that  the  thread  is  round  and  homogeneous,  it  can  be  made  to 
turn  rather  rapidly  on  itself,  in  one  sense  or  another,  without  appreciably 
affecting  the  plane  of  oscillation.  So  that,  at  the  pole,  the  experiment 
must  succeed  in  all  its  purity.  "  1 

"  But  when  our  latitudes  are  approached,  the  phenomenon  becomes 
complicated  in  a  way  that  is  rather  difficult  to  appreciate.  To  the 
extent  that  the  Equator  is  approached,  the  plane  of  the  horizon  has 
a  more  and  more  oblique  direction  with  respect  to  the  Earth.  The 
vertical,  instead  of  turning  on  itself  as  at  the  pole,  describes  a  cone 
which  is  more  and  more  obtuse. 

"  From  this  results  a  slowing  down  in  the  relative  motion  of  the 
plane  of  oscillation.  This  becomes  zero  at  the  Equator  and  changes 
its  sense  in  the  other  hemisphere.  " 

Without  explicitly  justifying  the  fact  in  his  paper,  Foucault  assumed 
that  the  angular  displacement  of  the  plane  of  oscillation  must  be  equal 
to  the  product  of  the  angular  motion  of  the  Earth  in  the  same  time 
with  the  sine  of  the  latitude.  If  the  correspondence  published  in  the 
collection  of  his  works  is  studied  in  this  connection,  it  is  apparent 
that  Foucault  arrived  at  this  relation  semi-intuitively,  before  it  had 
been  obtained  by  calculations  in  mechanics. 

At  first  Foucault  worked  on  a  relatively  modest  scale  by  suspending 
a  sphere  of  5  kg.  from  a  steel  wire  two  metres  long.  The  point  of 
support  was  a  strong  piece  of  casting  fixed  to  the  top  of  the  roof  of 
a  cellar.  He  took  the  precautions  of  ridding  the  wire  of  torsion  and 
ensuring  that  there  was  no  torsional  oscillation  of  the  sphere.  He 
"  encircled  the  sphere  with  a  loop  of  organic  thread  whose  end  is  attached 
to  a  point  fixed  on  the  wall,  and  chosen  so  that  the  oscillation  of  the 
pendulum  might  be  15  to  20°.  "  He  then  burnt  the  organic  thread. 
This  is  what  he  observed. 

"  The  pendulum,  subject  to  the  force  of  gravity  alone,  sets  off 
and  provides  a  long  sequence  of  oscillations  whose  plane  is  not  slow 
to  demonstrate  an  appreciable  displacement.  At  the  end  of  half  an 
hour  the  displacement  is  such  that  it  is  immediately  obvious.  But  it 

1  In  another  place  FOUCAULT  reassured  himself,  more  objectively,  that  "  whether 
or  not  the  Earth,  turning,  draws  along  the  point  of  attachment  with  the  monument 
[where  the  experiment  was  performed],  the  thread  experiences  no  torsion.  This 
implies  that  the  hob  of  the  pendulum  submits  to  this  motion  without  dragging  the 
plane  of  oscillation. " 


382  THE   EVOLUTION   OF   CLASSICAL 

is  more  interesting  to  follow  the  phenomenon  closely,  so  as  to  be  assured 
of  the  continuity  of  the  effect.  For  this  purpose  a  vertical  point,  con 
sisting  of  a  kind  of  style  mounted  on  a  support  placed  on  the  earth 
is  fixed  so  that  the  appendicular  projection  of  the  pendulum,  in  its 
to  and  fro  motion,  grazes  the  fixed  point  when  it  comes  to  its  extremitv. 
In  less  than  a  minute,  the  exact  coincidence  of  the  two  points  ceases. 
The  oscillating  point  is  continuously  displaced  towards  the  observer's 
left,  which  indicates  that  the  deviation  of  the  plane  of  oscillation  takes 
place  in  the  same  sense  as  the  apparent  motion  of  the  celestial  sphere. . . . 
In  our  latitudes  the  horizontal  trace  of  the  plane  of  oscillation  does 
not  complete  a  whole  circuit  in  twenty-four  hours.  " 

The  liveliness  and  the  accuracy  of  this  account  will  be  admired. 
As  we  have  indicated,  Foucault  had  started  his  work  in  a  cellar.  Thanks 
to  Arago,  who  put  at  his  disposal  the  meridian  room  at  the  Observatoire 
(Paris),  he  was  later  able  to  repeat  his  experiment  with  a  pendulum 
11  m.  long.  This  provided  a  slower  and  more  extensive  oscillation. 
Finally,  Foucault  worked  at  the  Pantheon  (Paris)  with  a  pendulum 
weighing  28  kg.  suspended  on  a  steel  wire  67  m,  long. 

As  Foucault  remarked — and  this  is  an  example  of  his  remarkable 
intuition — "  the  pendulum  has  the  advantage  of  accumulating  the  effects  * 
and  carrying  them  from  the  field  of  theory  into  that  of  observation.  " 

At  this  point  Foucault  referred  to  a  paper  of  Poisson,2  in  which 
the  latter  has  studied  the  deviation  of  projectiles. 

In  the  world  of  learning  Foucault's  experiment  had  the  immediate 
success  that  it  deserved.  Notes  accumulated  in  the  Comptes  rendus 
on  the  subject  of  the  pendulum  which  had  been  revived  in  this  way ; 
they  included  contributions  from  Binet,  Sturm,  Poncelet,  Plana, 
Bravais,  Quet,  Dumas,  etc. .  .  . 

Nevertheless,  however  brilliant  it  may  have  been,  Foucault's  expe 
riment  remained  rather  mysterious  to  the  general  public,  since  it  depend 
ed  on  the  displacement  of  a  plane  of  oscillation.  Moreover,  Foucault 
wished  to  give  a  still  more  tangible  proof  of  the  rotation  of  the  Earth. 

The  gyroscope  provided  him  with  a  means  of  doing  this.  He  used 
a  pendulum  suspended  by  its  centre  of  gravity  and  executing  what 
is  called  in  mechanics  a  motion  a  la  Poinsot. 

Foucault's  gyroscope  was  a  bronze  fly-wheel  mounted  inside  a 
metallic  circle  whose  diameter  contained  a  steel  axis  supporting  the 
wheel.  The  gyroscope  turned  about  one  of  its  central  axes  of  inertia, 
which  remained  fixed  in  space. 

1  Without  this  accumulation  FOUCAULT  would  not  have  been  able  to  detect  a  force 
that  was  only  the  55,000th  part  of  the  weight  of  the  Pantheon  pendulum. 

2  Comptes  rendus  de  VAcad^mie  des  Sciences,  November  13th,  1837. 


RELATIVE   MOTION  383 

Foucault  wrote1  "The  body  can  no  longer  participate  in  the  diurnal 
motion  which  actuates  our  sphere.2  Indeed,  although  because  of  its 
short  length,  its  axis  appears  to  preserve  its  original  direction  relatively 
to  terrestrial  objects,  the  use  of  a  microscope  is  sufficient  to  establish 
an  apparent  and  continuous  motion  which  follows  the  motion  of  the 
celestial  sphere  exactly.  .  .  .  As  the  original  direction  of  this  axis 
is  disposed  arbitrarily  in  all  azimuths  about  the  vertical,  the  observed 
deviations  can  be,  at  will,  given  all  the  values  contained  between  that 
of  the  total  deviation  and  that  of  this  total  deviation  as  reduced  by 
the  sine  of  the  latitude.  " 

Foucault  concludes,  in  a  somewhat  journalistic  style  that  was 
probably  natural  to  the  reporter  of  the  Debats — 

"  In  one  fell  swoop,  with  a  deviation  in  the  desired  direction,  a 
new  proof  of  the  rotation  of  the  Earth  is  obtained;  this  with  an 
instrument  reduced  to  small  dimensions,  easily  transportable,  and 
which  mirrors  the  continuous  motion  of  the  Earth  itself.  ...  In  your 
possession  are  pieces  of  material  which  are  truly  subject  to  the  dragging 
of  the  diurnal  motion.  " 

Thus   Fourier  achieved  one   of  Poinsot's   aims. 

The  compound  centrifugal  force  in  the  sense  of  Coriolis,  and 
Foucault's  pendulum,  are  two  essential  achievements  in  mechanics  ;  the 
one  has  an  origin  which  is  purely  mathematical,  the  other  was  the 
product  of  a  physicist's  brilliant  intuition.  Though  they  are  united 
in  the  same  rational  exposition  in  the  books  that  are  now  classical,  they 
were  born  separately — it  was  not  the  reading  of  Coriolis  that  inspired 
Foucault's  experiment. 

1  Comptes  rendus  de  VAcademie  des  Sciences,  Vol.  35,  p.  421   (September    27th, 
1852). 

2  More  correctly,  it  is  easy  to  give  the  gyroscope  a  very  rapid  proper  rotation  about 
its  own  axis,  say  o>,  which  is  very  large  compared  with  the  absolute  rotation  of  the 
Earth.,  say  Q. 

If  Q  is  the  absolute  rotation  of  the  gyroscope, 

Q  =  co  +  Q 

and  the  axis  remains  directed  towards  the  fixed  stars  (U  co  co)  as  long  as  Q  is  negligible 
compared  with  co. 


CHAPTER   FIVE 


POISSON'S  THEOREM  (1809) 


1.    POISSON'S    THEOREM    AND    BRACKETS. 

Poisson's  theorem  appeared  among  the  investigations  made  im 
mediately  after  the  appearance  of  two  papers  by  Lagrange.  One  of 
these  papers  appeared  in  1808  and  the  other  in  1809,  and  they 
were  incorporated  in  the  1811  edition  of  the  Mecanique  analytique. 
Stimulated  by  the  needs  of  the  theory  of  perturbations  in  classical 
mechanics,  they  were  concerned  with  the  variation  of  arbitrary  cons 
tants. 

Here  we  shall  follow  a  paper  of  Poisson  which  was  read  at  the 
Institut  de  France  on  October  16th,  1809.1 

Poisson  starts  from  Lagrange's  equations 


Putting  R  =  T  —  V,  where   V  depends  only  on  the  ql  and  not   on 
the  y  »,  he  obtains 

dR       dT 

5-7  =  5-7  =  ut 
dqi        dql 

whence 

du1_dR 

W  dt    ~  d^ 

66  In  this  way  the  equations  of  motion  are  reduced  to  the  simplest 
form  that  they  can  be  given.  " 

1  Journal  de  MZcole  polytechnique,  cahier  XV,  1809,  p.  266. 

While  following  the  development  of  POISSON'S  analysis  rigorously,  we  have  taken  the 
liberty  of  condensing  its  form  by  using  the  convention  of  the  summation  of  dummy 
suffixes.  This  is  commonly  used  in  the  absolute  differential  calculus  and  allows  the 
direct  consideration  of  a  system  of  k  degrees  of  freedom  (rather  than  three,  as  POISSON 
did).  Further,  we  have  introduced  the  distinction  between  the  symbols  of  partial  and 
total  derivatives. 


POISSON  385 

The  new  variables  ut  are  functions  of  the  q,,  and  the  gj.  Con 
versely,  the  q'i  can  be  regarded  as  functions  of  the  ql  and  the  ut. 

Considering  R  as  a  function  of  the  qt  and  the  ut,  Poisson  denotes 
the  partial  derivatives  of  jR  when  the  independent  variables  are  ql  and 

8R 

Ui  by  -  —  ;  and  the  derivatives  of  J?  when  the  independent  variables 

vqi  ,  df?\ 

are  q^  and  q't  by  (  —  ).     Thus  equation  (1)  becomes 


dut  _  /dR 
~dt"~  \dq 

However 


(  '  dqt         dq 

so  that 

dui       BR 


Tke  partial  derivatives  of  R  may  be  written 


Finally,  by  calculating  -  -  in  two  different  ways,  it  is  found  that 

OUj,  OUj 

2    =  Uk  ^   |  w 

dUiduj       dul 
92q'k      ,     Bqi 

-  -  --  J  --  =-  . 


From  this  is  obtained  the  relation 


which  will  be  used  in  the  sequel. 

Given  this,  Poisson  considers  a  first  integral  of  the  equations  of 
motion  containing  a  single  arbitrary  constant  a.  This  integral  equation, 
if  solved  for  a,  would  lead  to 

a  =  funct  (qi  ...  q^  Ui  .  .  .  u&,  t). 

25 


386  THE   EVOLUTION    OF   CLASSICAL 

Hence 

.        3a   ,    da    ,   .    da     , 

0  =  ar 

da 


da    ,        da  fdR  _       dq'r\ 
^kqk+^k  \dqk~~  Ur  3qJ  ' 


Differentiating  with  respect  to  q^ 
«rt  n  _  L  i^\  +  ^  ^  +  -^ 

W  dt   \~dqj  "*"  Sqk  dqt  "*"  8uk 

If  another  first  integral  of  the  equations  of  motion,  containing 
an  arbitrary  constant  6,  had  been  considered,  then 

m  o  -  A  ffl  4-  1*  ^  +  ^-  f  82jR  -  « 

l/  j  <ft   \ag/  "^  a9&  dq>  ^  duk  T 


By  multiplying  equation  (6)  by  —  and  equation  (7)  by  —  —  ,  sum- 

OUt  OUi 

ming  over  the  dummy  suffixes  and  adding  the  two  equations, 

0  -  —  ™  (—]  —  ——  (—}  4-^(—  —  —  ——} 
^  '          ""  dui  dt  \dqi/       dut  dt  \dqi/        dqi  \dqk  dui        dqk  duj' 

Differentiating  the   equation  from  which   (6)   was   obtained  with 
respect  to  ut,  instead  of  q^  there  is  obtained 


_  u 
r 


dt  \3tti /        dqk  3ut       duk  \dqkdui  dqkdqi/        duk  dqk 

The  third  term  vanishes  because  of  (4).     There  remains 

(£tt\  t\        d  1 3^\        ^a  dqic        da  dqi 

dt  \duj        dqk  dui       duk  dqk' 

Also,  for  the  other  first  integral, 

dt  \diii/       dqk  dui       duk  dqk 

Multiplying  (6")   by  —  —  and  (7")  by  — ,    summing    over    the 
dummy  suffixes  and  adding  the  two  equations, 


.----        - 
dt  \du        dqi        dqi  dt  \du 

dqL  idb_  9a  _  db_  9a  \  dqi  idb_  da_  _  jtt    da_ \ 

dui  \dqk  dqi        dqi  dqj  dqk  \duk  dqi        dqi  duk/  ' 


POISSON  387 


The  third  term  of  this  equation  is  zero  because  of  the  relation  (5). 
If  the  suffixes  i  and  k  are  interchanged  in  the  fourth  term,  by  adding 
(8)  to  (8')  it  follows  that 


y    d  (da\        da    d^  fdb\        da^  d_  /db\  __  db_   d_  ida\  _ 
~i  di  (dqj  ~"faidt  \dq~J  +  dq'i  dt  \duj       dqi  dt  \duj  ~~ 


db   d 
du 


or 


d  [  db    da        da    db  \        A 

_ _ —  _ —     =  u 


dt 

or  finally 

db    da        db    da 

—  -  --  —  —  =  constant 

dui  dqi        dqt  dut 

or 

(fe9  a)  =  constant 

where  (i,  a)  is  an  expression  which  has  become  known  as  a  "  Poisson 
bracket.  " 

It  is  evident  that 

(6,  a)  =  —  (a,  b)  and  that  (a,  a)  =  0. 

Poisson  concludes,  "  The  analysis  that  we  have  just  performed 
therefore  leads  us  to  this  remarkable  result  —  that  if  the  values  of 
the  arbitrary  constants  on  the  integrals  of  the  equations  of  motion 
of  a  system  of  bodies  are  expressed  as  functions  of  the  independent 
variables  (q,)  and  the  quantities  (ut),  the  combination  of  the  partial 
differentials  of  these  functions  that  is  represented  by  (a,  b)  will  always 
be  a  constant  quantity.  " 

This  proposition,  which  has  become  classical,  evidently  exhibits 
considerable  aesthetic  value.  Its  practical  content  is  more  limited. 
Indeed,  Poisson's  theorem  seems  to  indicate  that  it  is  sufficient  to 
know  two  first  integrals  of  the  equations  of  motion  in  order  to  be 
able  to  deduce  a  third  from  them  ;  by  combining  this  with  one  of  the 
first  two,  a  fourth  would  be  obtained,  etc  ----  But  if  the  bracket 
(a,  b)  is  identically  constant,  or  if  it  is  a  function  of  the  integrals  already 
known,  this  process  contains  nothing  new. 

2.    THE    LAGRANGE-POISSON    SQUARE    BRACKETS. 

Poisson  supposes  that,  to  the  right  hand  side  of  the  Lagrange  equa 
tions 


dt  \Sq' 


388  THE   EVOLUTION   OF   CLASSICAL 

is  added  a  term  depending  on  the  function  of  the  perturbing  forces, 
Q.    This  yields 


Since  the  variables  u£  are  always  defined  by  —  ,    Lagrange's    equa- 
tions  take  the  form 


The    expression    I  — )    must   always    be   interpreted    as    a    derivative 

in  which  the  qi  and  the  q[  are  chosen  as  the  independent  variables. 
This  distinction  does  not  apply  to  derivatives  of  £?,  which  is  supposed 
to  be  independent  of  the  g£. 

If  the  equations  (1)  are  integrated  completely,  so  that  the  solution 
contains  2k  arbitrary  constants  a,,  it  is  desired  to  satisfy  the  equations 
(2)  by  varying  the  arbitrary  constants.  Since  the  number  of  these 
is  twice  the  number  of  the  equations  (2),  the  2k  quantities  as  can  be 
restricted  by  any  k  conditions  that  may  be  chosen. 

Poisson  supposed,  "  as  in  the  theory  of  the  Planets  "  that  the  dif 
ferentials  of  the  variables  gf  kept  the  same  form  independently  of 
whether  the  as  were  constants  or  not. 

He  goes  on  to  express  this  condition  by  the  k  relations 

tA\  *          d^  i          A  (£  =  1,  2  . ..  k) 

(4)  dqt  =  ^-  das  =  0  }  / 

3        das  (s  from  1  to  2k). 

On  the  other  hand 

^  dt  +  ™  *. 


But  when  the  a,  are  constants,  the  first  two  terms  are  equal  because 
of  (1).  Accordingly,  if  only  the  as  vary 

(5)  du^^-dt. 

oqi 

T,  j|  The  2k  equations  (4)  and  (5),  which  are  linear  and  of  the  first  order 
in  the  quantities  da,,  determine  the  differentials  of  the  arbitrary  cons 
tants. 

Let  ar  =  funct  (t,  31,  ...  qk,  MI,  ...  uk) 

be  a  first  integral  of  the  equations  (1).     Then  put 

,          dar  dQ  _ 
dar  =  — -  -—  dt . 


POISSON  389 

Now  X*_  =  ?R.fa. 

dqj        das     dqj 

Therefore 

,          dar  das  dQ  _  /j  from  1  to  k  \ 

^  *  T       duj  dqj  das  \s  from  1  to  2k' 

But  in 


the  derivatives    —  are  necessarily  zero.     Therefore 

3Uj 

0  =  —  =  —  ^i 
duj       3as  duj 

or,  by  slimming  the  equations  (7)  after  multiplying  them  by  —  -, 

dqj 

(8)  Q==3Qda.dar 

das  duj  dqj' 

By  subtracting  (8)  from  (6) 

7          dQ  .  (dar  das       das  dar\ 


and,  using  the  definition  of  the  Poisson  brackets, 

,         SQ  .   .         .  fr  =  1,  2  .  .  .  2Jfc\ 

Jar  ==  —  -  dt  (ar,  as)  (  I  . 

das  \s  from  1  to  2k  I 

The  brackets  (ar,  as)  are  functions  only  of  the  arbitrary  constants 
a-L,  .  .  .  .  ,  a2fc.  "  It  follows  that,  in  the  equations  of  mechanics,  the 
first  differentials  of  the  arbitrary  constants  can  be  expressed  by  means 
of  the  partial  differences  of  the  function  Q,  taken  with  respect  to  these 
quantities  and  multiplied  by  functions  of  these  same  quantities,  which 
do  not  contain  the  time  explicitly.  This  is  the  beautiful  theorem 
that  Mr.  Lagrange  and  Mr.  Laplace  first  discovered  in  connection 
with  the  differences  of  elliptic  elements,  and  which  Mr.  Lagrange  then 
extended  to  a  system  of  any  bodies  subject  to  forces  directed  towards 
fixed  or  movable  centres  and  whose  intensities  are  functions  of  the 
distances  of  the  bodies  from  these  centres.  " 

Lagrange  had  arrived  at  the  formulae 

-—dt=  [ar,  aj  das 
dar 

in  which  the  square-bracket  expression  had  the  value 

!-„  ^i_?2i^_M^ 

Lar'asJ~aar  das       BasdOr' 


CHAPTER    SIX 

ANALYTICAL  DYNAMICS 
IN  THE  SENSE  OF  HAMILTON  AND  JACOBI 


1.  HAMILTON   OPTICS.    ITS    DOUBLE    INTERPRETATION   IN   TERMS    OF 
EMISSION  AND  WAVE  PROPAGATION. 

Hamilton's  ideas  on  dynamics  cannot  be  divorced  from  his  ideas 
on  optics.  For  this  reason  it  is  essential  that  we  should,  for  a  few 
moments,  concern  ourselves  with  the  latter. 

We  shall  follow  the  edition  of  Hamilton's  works  that  has  been 
published  by  the  Royal  Academy  of  Ireland.  Apart  from  the  papers 
which  have  been  known  and  classical  for  some  time,  this  edition, 
very  fortunately,  contains  extracts  from  the  numerous  note-books 
which  Hamilton  kept  and  which  had  not  been  published  before.  No 
doubt  the  author,  considering  them  minor  works,  had  not  wished  to 
make  them  public — but  they  throw  the  work  of  this  inspired  Irishman 
into  a  new  and  very  interesting  light.1 

In  the  first  place  we  shall  cite  an  article  which  appeared  in  the 
Dublin  University  Review 2  for  October,  1833,  called  On  a  general 
method  of  expressing  the  paths  of  light,  and  of  the  planets,  by  the  coeffi 
cients  of  a  characteristic  function. 

At  the  time  that  Hamilton  started  his  investigations  in  optics, 
neither  the  theory  of  waves  nor  the  emission  theory  were  generally 
accepted.  Hamilton's  geometric  optics,  which  was  essentially  a  new 
method  of  formalising  the  coUection  of  results  that  had  already  been 
obtained,  was  capable  of  being  interpreted  in  terms  of  wave  propagation 
(in  Huyghens*  sense)  and  corpuscles  (in  the  sense  of  the  dynamical 
principle  of  least  action).  This  was  the  essential  merit  of  his  theory, 

1  The  Mathematical  Papers  of  Sir  William  Rowan  Hamilton.     Vol.  I,  Geometrical 
Optics  edited  for  the  Royal  Irish  Academy  by  A.  W.  CONWAY  and  J.  L.  SYNGE  (1931) ; 
Vol.  II,  Dynamics  edited  for  the  R.  I.  A.  by  A.  W.  CONWAY  and  J.  McCoNNEL  (1940). 
Cambridge  University  Press. 

2  Pp.  795-826. 


HAMILTON    AND    JACOBI  391 

which,  can  only  seem  more  meritorious  to  our  modern  age  in  which 
a  similar  dualism  has  been  established  in  theoretical  physics. 

A  great  admirer  of  Lagrange,  Hamilton  declared,  in  the  article 
referred  to  above,  that  he  was  "  struck  by  the  imperfection  of  deductive 
mathematical  optics.  "  He  wished  to  give  to  optics,  on  the  plane  of 
formal  theory,  the  same  "  beauty,  power  and  harmony  "  with  which 
Lagrange  had  been  able  to  endow  mechanics. 

I  repeat  that  it  was  certainly  the  formalism  which  concerned  him. 
"  Whether  we  adopt  the  Newtonian  or  the  Huyghenian,  or  any  other 
physical  theory,  for  the  explanation  of  the  laws  that  regulate  the 
lines  of  luminous  or  visual  communication,  we  may  regard  these  laws 
themselves,  and  the  properties  and  relations  of  these  linear  paths 
of  light,  as  an  important  separate  study,  and  as  constituting  a  separate 
science,  called  often  mathematical  optics. " l 

Hamilton  recalled  the  development  which  we  have  already  studied, 
Fermat,  Maupertuis,  Eider,  Lagrange.  w  But  although  the  law  of  least 
action  has  thus  attained  a  rank  among  the  highest  theorems  of  physics, 
yet  its  pretensions  to  a  cosmological  necessity,  on  the  ground  of  economy 
in  the  universe,  are  now  generally  rejected.  And  the  rejection  appears 
just,  for  this,  among  other  reasons,  that  the  quantity  pretended  to  be 
economised  is  in  fact  often  lavishly  expended.  "  This,  for  instance,  is 
what  is  shown  in  the  commonplace  case  of  reflexion  on  a  spherical 
mirror,  where  obviously  if  one  of  the  rays  issuing  from  a  point  is 
minimal,  the  other  corresponds  in  fact  to  a  maximal.  We  can  therefore 
speak  reasonably  only  of  a  stationary  property  of  the  action  (or  an 
extremal  one,  as  understood  in  the  calculus  of  variations). 

"  We  cannot,  therefore,  suppose  the  economy  of  this  quantity  to 
have  been  designed  in  the  divine  idea  of  the  universe :  though  a  simplic 
ity  of  some  high  kind  may  be  believed  to  be  included  in  that  idea  **. 

Such  are  the  rational  motives  which  led  Hamilton,  at  the  same  time 
as  he  retained  the  consecrated  term  action,  to  speak,  in  optics  as  in 
dynamics,  of  stationary  or  varying  action,  according  to  whether  the 
extremities  of  the  rays  or  trajectories  are  fixed  by  hypothesis  or  not. 

I  shall  pass  over  the  remarkable  statement  of  the  principles  of  the 
calculus  of  variations  contained  in  the  paper  we  are  analysing,  and  come 

1  Incidentally  HAMILTON  did  not  hesitate  to  state  his  doctrine  of  scientific  philo 
sophy.  Thus  he  distinguished  a  stage  in  which  the  facts  are  raised  to  laws  by  induction 
and  analysis,  and  another  in  which  the  laws  are  used  to  obtain  consequences  by  deduction 
and  synthesis.  This  was  formulated  in  the  following  remarkable  passage. 

**  We  must  gather  and  group  appearances,  until  the  scientific  imagination  discerns 
their  hidden  laws,  and  unity  arises  from  variety  ;  and  then  from  unity  we  must  rede- 
duce  variety,  and  force  the  discovered  law  to  utter  its  revelations  of  the  future.  " 

Better  than  a  fine  formula,  this  thesis  is  the  expression  of  the  method  of  work  that 
HAMILTON  always  followed. 


392  THE   EVOLUTION   OF   CLASSICAL 

to  the  exact  statement  of  the  Hamiltonian  principle  of  stationary  action 
in  optics  : 

"  The  optical  quantity  called  action,  for  any  luminous  path  having  i 
points  of  sudden  bending  by  reflexion  or  refraction,  and  having  therefore 
i  +  1  separate  branches,  is  the  sum  of  i  +  1  separate  integrals, 

ACTION  =  V  =  vjrfl/i  =  Fi  +  F2  +  . . .  +  Vl+l 
of  ichich  each  is  determined  by  an  equation  of  the  form 


the  coefficient  vt  of  the  element  of  the  path,  in  the  ith  medium,  depending, 
in  the  most  general  case,  on  the  optical  properties  of  that  medium,  and 
on  the  position,  direction  and  colour  of  the  element,  according  to  rules 
discovered  by  experience.  (For  example,  if  the  ith  medium  is  an  ordinary 
medium,  vt  is  its  refractive  index.)  This  quantity  V  is  stationary 
in  the  propagation  of  the  light.  " 

The  law  of  varying  action  is  a  generalisation  of  the  stationary  law 
in  which  the  ends  of  an  optical  (luminous)  path  are  allowed  to  vary. 
The  conditions  at  the  limits  thus  make  necessary  the  intervention 
of  finite  difference  equations  of  the  type  A  V  =  Xu  =  0  on  each  surface 
u  =  0  of  reflection  or  refraction.  In  passing,  Hamilton  indicated 
that  the  remarkable  permanence  of  what  he  called  the  components 
of  normal  slowness  (inversely  proportional  to  those  of  the  velocity 
of  wave  propagation  in  Huyghens'  sense)  had  been  suggested  to  him 
by  the  observation  that  the  characteristic  function  V  is  such  that 
the  wave  surfaces  satisfy  the  equation 

V  =  Constant. 

The  components  of  normal  slowness  are  nothing  else  than  the  partial 

,    .       .       dV  dV  dV 

derivatives  — ,  — *  -5--     Thus    is    rediscovered    the    theorem — then 

ox     oy    oz 

disputed — of  Huyghens  according  to  which  the  rays  of  every  homo 
geneous  system,  starting  from  a  single  point  or  normal  to  a  surface, 
remain  normal  to  a  family  of  surfaces  after  they  have  been  subjected 
to  any  number  of  reflections  or  refractions. 

We  learn  a  little  more  about  Hamilton's  procedures  in  optics  by 
following  the  Third  Supplement  to  an  Essay  on  the  Theory  of  Systems 
of  Rays. 

First,  calling  the  initial  and  final  coordinates  of  a  ray  (x',yr,  zf)  and 
(x,  y,  z),  Hamilton  writes 

<A)       *-+»  +      »-*--« 


HAMILTON   AND   JACOBI  393 

(where  a,  /?,  y  and  a',  /?',  y'  are  the  direction  cosines  of  the  ray  at  its 
end    and    beginning    respectively)    or 


On  the  other  hand,  the  conditions  for  an  extremum  require  that 


w  E*-*E--   E*-'S  <* 

The  function  t;  is  homogeneous  in  a,  /?  and  7  and  also  depends  on  the 
frequency  of  the  light.  Hamilton  expresses  the  latter  fact  by  the 
introduction  of  a  chromatic  index  %. 

Thus  he  arrives  at  the  following  two  equations  in  the  first  partial 
derivatives 


SV   3V  . 


/-SV  —dV  -SV 


(C) 


The  similarity  of  the  form  of  these  equations  with  that  of  the 
equations  of  dynamics  is  evident —  V  corresponds  to  the  action  integral 
(in  the  Euler-Lagrange  sense)  ;  the  equation  (C)  corresponds  to  the 
equation  of  kinetic  energy  and  %  to  a  certain  function  of  the  total 
energy.  Moreover,  Hamilton's  optical  equations  can  be  easily  written 
in  the  canonical  form  that  he  himself  gave  to  the  equations  of  dynamics. 
It  is  sufficient  to  denote  the  components  of  normal  slowness  (or  the  partial 
derivatives  of  V)  by  cy,  r  and  v  to  write 

dx  ___  9flj  da  __       dQ 

•  7  Tjr  ~^ 1  •    •    •  ~7~T>  ^ etC. 

dV       do*  dV  dx 

As  we  have  already  indicated,  Hamilton  interpreted  the  action 
F,  in  the  language  of  the  wave  theory,  as  the  time  necessary  for  a 
wave  of  frequency  %  which  starts  from  the  point  (x' ',  y',  2')  to  travel 
to  the  point  (#,  y,  z). 

If  the  wave  velocity  (ondulatory  velocity)  of  propagation  along 
the  corresponding  radius  is  called  u,  the  relation 

(D)  u  -  1 

v     /  v 

or,  more  generally, 


394  THE   EVOLUTION   OF   CLASSICAL 

allows  V  to  be  written  as 

F=J"V"          u      (g         g  *  y        ,  y 

Since  the  rays  are  identical  in  the  two  theories,  to   the   extrema 

dV=  dfvds  =  0 
of  the  ^mission  theory  there  corresponds  the  extremal 


which  is    Fermat's  principle. 

There  is,  here,  the  germ  of  the  transcription  which  Schrodinger 
was  to  turn  to  good  use  in  dynamics,  in  generalising  equation  (D)  by 
the  introduction  of  a  group  velocity  (in  Rayleigh's  sense)  identified 
with  v. 

2.    THE   DYNAMICAL    LAW    OP   VARYING   ACTION   IN    HAMILTON'S    SENSE. 

Historically,  Hamilton's  first  work  in  dynamics  is  contained  in 
a  manuscript  dated  1833  and  called  The  Problem  of  Three  Bodies  by 
my  Characteristic  Function.3-  He  treated  the  problem  of  the  Sun, 
Jupiter  and  Saturn  and  introduced,  from  the  beginning,  the  charac 
teristic  function. 

V=  f  2Tdt. 

Jo 

(The  living  force  accumulated  from  the  origin  of  time  to  the  time 
t.)  Hamilton  showed  that  this  function  must  satisfy  two  equations  in 
the  partial  derivatives  of  the  first  order.  He  then  compared  an  approx 
imate  solution  of  this  problem  with  that  obtained  by  Laplace,  studied 
the  perturbations,  determined  the  characteristic  function  of  elliptical 
motion  and  established  the  equation 

3V 

—  =zz  t       (A,  constant  of  living  forces) . 

He  then  proved  that  the  two  equations  connecting  the  partial  derivatives 
of  V  have  a  common  solution,  and  directed  his  attention  to  the  deter 
mination  of  this  solution  by  successive  approximations. 

Therefore  this  paper  already  contained  essential  results.  We 
shall  not,  however,  further  discuss  it,  for  Hamilton  undertook  the 
codification  of  these  investigations  in  two  fundamental  papers  which 
were  published  in  1834.  We  propose  to  analyse  these. 

1  Note-book  29. 


HAMILTON   AND    JACOBI  395 

In  his  statement  of  the  intentions  of  his  First  Essay  on  a  General 
Method  in  Dynamics  l  Hamilton  recalled  that  the  determination 
of  the  motion  of  a  system  of  free  particles,  subject  only  to  their 
mutual  attraction  or  repulsion,  depended  on  the  integration  of  a  system 
of  3  (n  —  1)  ordinary  differential  e<juations  of  the  second  order  or, 
by  a  transformation  due  to  Lagrange,  on  a  system  of  6(n  —  1)  ordinary 
differential  equations  of  the  first  order. 

Hamilton  reduced  this  problem  to  the  "  search  and  differentiation 
of  a  single  function  "  which  satisfied  two  equations  of  the  first  order 
in  the  partial  derivatives. 

From  this  transference  of  the  difficulties,  even  if  it  is  thought 
that  no  practical  advantage  results,  "  an  intellectual  pleasure  may 
result  from  the  reduction  of  the  most  complex  and,  probably,  of  all 
researches  respecting  the  forces  and  motions  of  bodies,  to  the  study 
of  one  characteristic  function,  the  unfolding  of  one  central  relation.  ..." 
And  Hamilton  adds,  "  this  dynamical  principle  is  only  another  form 
of  that  idea  which  has  already  been  applied  to  optics  in  the  Theory 
of  systems  of  rays.  .  .  .  " 

Starting  from  the  classical  equation 

(1)          £  m  (*"<5*  +  y%  +  ****)  =  <5U"      with       17  =  £  mm'/(r), 


in  which  U  is  the  function  of  forces,  Hamilton  denotes  the  living  force 
of  the  system  by  2T  =  E  ™  (*'2  +  /2  +  *'2)>  and  writes  the  law 
of  living  forces  in  the  form 

T  =  U  +  H. 

The  quantity  H  —  which  it  has  become  customary  to  call  the  Hamil- 
tonian  of  the  system  —  is  independent  of  the  time  in  a  given  motion 
of  the  system.  But  when  the  initial  conditions  are  varied,  H  varies 
correspondingly  according  to  the  equation 

dT  =  SU  +  6H. 

On  multiplying  by  df,  integrating  from  0  to  f,  using  equation  (1) 
and  the  equation  that  defines  the  kinetic  energy,  Hamilton  obtains 


J^  2  m  (dxM  +  dydy'  +  <W)  =  J"o  £  m  (dx'dx  +  dy'dy  +  dz'dz)  + 

Then,  by  means  of  the  calculus  of  variations 

(A)       6  V=  £  m  (x'dx  +  y'dy  +  z'dz)  —  £  m  (a'da  +  b'db  +  c'8c)  +  tdH 

1  Phil  Trans.  Roy.  Soc.  (1834),  II,  p.  247. 


396 


THE   EVOLUTION   OF  CLASSICAL 


where  (#,  y,  z)  and  (a,  &,  c)  are  the  final  and  initial  coordinates  of  the 
points  of  the  system,  and  V  is  the  function 

(B)  v = Jo  S  m  (x'dx + y'Jy  +